User:RJGray/Sandboxcantor

To work on: https://www.insurance.com/health-insurance/health-insurance-basics/open-enrollment-what-you-need-to-know.html

Misc

https://www.bbc.co.uk/bitesize/guides/zg8pycw https://www.bbc.co.uk/bitesize/guides/zg8pycw/revision/4

Cantor at FAC (part 2)

Hi Iry-Hor, I was wondering if you have the time to do some more mentoring work for "Georg Cantor's first set theory article". I made quite a few improvements since last January.

The biggest one was rewriting the section "The disagreement about Cantor's existence proof." The new section is: "A misconception about Cantor's work". A set theorist didn't like my "disagreement" section. Since I did not want to nominate the article for FA knowing that there was a set theorist who objected to one of its sections. So I discussed the disagreement section with him until I figured out what bothered him. Then, I searched the literature for an approach that probably would receive no complaints. I finally found it in an article by the set theorist Akihiro Kanamori.

I also made the article more accessible to people who are blind by installing some software that allowed me to hear the mathematics in the article. It turned out that some of my original code was confusing for a blind person to listen to. For example, in the Second Theorem section, my original code was confusing because the code for the 3 illustrations came after the code for the 3 descriptions. I fixed this by making each of the 3 descriptions start with its illustration and finish with its description. The new code produces the same text and illustrations as the old code.

With these changes and several others, I feel this article is ready for the final steps in the featured article process.

Hard to understand exactly what the featured article wants--the directions seem a bit scattered without an example.

To work on: (Featured article): #Featured article info

To work on: (Featured article): Cantor's first set theory article

To work on: (Featured article): WP:MOS WP:FA

To work on: (Featured article): User talk:RJGray#Cantor draft

To work on: (Featured article): User talk:RJGray#Cantor at FAC

To work on: (Featured article): User_talk:Iry-Hor/Archive_11#Cantor_at_FAC

Locke, An Essay concerning Human Understanding

CHAP. XV, §12, p. 188. What I say of man I say of all finite beings; who, though they may far exceed man in knowledge and power, yet are no more than the meanest creature, in comparison with God himself. Finite of any magnitude holds not any proportion to infinite. God’s infinite duration being accompanied with infinite knowledge and infinite power, he sees all things past and to come; and they are no more distant from his knowledge, no farther removed from his sight, than the present: they all lie under the same view; and there is nothing which he cannot make exist each moment he pleases.

CHAP. XVII.: Of Infinity, § 1, p. 194. It is true, that we cannot but be assured, that the great God, of whom and from whom are all things, is incomprehensibly infinite: but yet when we apply to that first and supreme being our idea of infinite, in our weak and narrow thoughts, we do it primarily in respect to his duration and ubiquity; and, I think, more figuratively to his power, wisdom, and goodness, and other attributes, which are properly inexhaustible and incomprehensible, &c. For, when we call them infinite, we have no other idea of this infinity, but what carries with it some reflection on, and imitation of, that number or extent of the acts or objects of God’s power, wisdom, and goodness, which can never be supposed so great or so many, which these attributes will not always surmount and exceed, let us multiply them in our thoughts as far as we can, with [195] all the infinity of endless number.

CHAP. XVII, § 13, p. 202. Though it be hard, I think, to find any one so absurd as to say, he has the positive idea of an actual infinite number; the infinity whereof lies only in a power still of adding any combination of units to any former number, and that as long and as much as one will; the like also being in the infinity of space and duration, which power leaves always to the mind room for endless additions; yet there be those who imagine they have positive ideas of infinite duration and space. It would, I think, be enough to destroy any such positive idea of infinite, to ask him that has it, whether he could add to it or no; which would easily show the mistake of such a positive idea. We can, I think, have no positive idea of any space or duration which is not made up, and commensurate to repeated numbers of feet or yards, or days and years, which are the common measures, whereof we have the ideas in our minds, and whereby we judge of the greatness of this sort of quantities. And therefore, since an infinite idea of space or duration must needs be made up of infinite parts, it can have no other infinity than that of number, capable still of farther addition: but not an actual positive idea of a number infinite. For, I think, it is evident that the addition of finite things together (as are all lengths, whereof we have the positive ideas) can never otherwise produce the idea of infinite, than as number does; which consisting of additions of finite units one to another, suggests the idea of infinite, only by a power we find we have of still increasing the sum, and adding more of the same kind, without coming one jot nearer the end of such progression.==

Featured article info

From: Editing Help:Wikipedia: The Missing Manual/Collaborating with other editors

Going for the gold: Better and best article candidates

Wikipedia has two classifications for high-quality articles that have been through an assessment nomination process: Good and Featured. Below are five places where assessments take place, and you may be able to contribute.

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If you're an academic, scientist, or engineer, or an expert in a particular field, then the articles at the pages listed in this section could particularly benefit from your comments, even if you're relatively new to Wikipedia.

Candidates for Good and even Featured classification may be a long way from perfect. You may find the checklist approach to improving articles described in Chapter 18: Better articles: A systematic approach a big help here. As always, when you're looking over listed articles, you can pick and choose. You don't have to comment on articles you're not interested in, or where you don't see obvious opportunities for improvement.

Wikipedia:Good article nominations (shortcut: WP:GAN). At any given time, you'll probably find several hundred articles undergoing review, nicely organized into topical categories.
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Wikipedia:Featured article candidates (shortcut: WP:FAC). Nominees for Wikipedia's highest quality category are on this page—usually 50 to 100 articles at a time. Articles are often up for a month or two while undergoing review, so checking in every 3 or 4 weeks to see what's up is frequent enough.

As with Good article nominations, Featured article candidates have almost always been nominated by the editors who created or significantly improved those articles. These editors are available, motivated, and capable of fixing just about anything that other editors identify as needing attention. If you make detailed suggestions, you may be gratified by quick responses to your comments.

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Wikipedia:Featured article review (shortcut: WP:FAR) is similar to good article reassessment. FAs do acquire problems and errors after they've passed their candidacy.

These reviews take place in two stages: First, a basic review with the goal of improving the article. Second, when improvements are inadequate, the article is declared a removal candidate, and editors declare whether they support keeping or removing the article's FA status; this stage is also an opportunity for editors to overcome deficiencies. Each stage typically lasts 2 to 3 weeks. Typically, a dozen or so articles are in each stage at any given time.

Wikipedia:Featured list candidates (shortcut: WP:FLC). Lists are a specialized type of article (see the section about list articles). Much of the discussion on this page is about formatting, particularly tables (Chapter 14: Creating lists and tables).

To work on: (Cantor's 1883 article): User:RJGray/Sandboxcantor1

Free ref for Mathematische Annalen. Backup if use of current one goes bad.

Cantor, Georg (1879), "Ueber unendliche, lineare Punktmannichfaltigkeiten. 1.", Mathematische Annalen (in German), 15: 1–7, doi:10.1007/bf01444101 {{citation}}: Invalid |ref=harv (help).

Fraenkel 1928: Dieser interessante und bestimmte Satz, der über die damals (1874) von den transzendenten Zahlen bekannten Tatsachen weit hinausgeht, betrifft eine Menge von Zahlen, bon denen auch nur eine einzige wirklich zu bestimmnen keineswegs ganz leich ist.

Fraenkel 1928: (Linguee) This interesting and definite proposition, which goes far beyond the facts known at that time (1874) about transcendental numbers, concerns a number of numbers for which it is by no means easy to determine even a single one.

Hi, Trovatore. I wish to congratulate you on your choice of a sentence to delete from the lead section of Cantor's first set theory article. Obviously, it's a high profile sentence that you disagree with. But it's also a sentence that has started to bother me: Putting dates of 2014 and 2015 in the lead section might be read as dating the research, which would be inaccurate and not relevant for a lead. However, I hadn't decided how to modify it. Your elimination of this sentence solved the problem for me. So, thank you. The other sentence you removed leads into the following sentence so it emphasizes what I am saying. I like that sentence more, but I don't think it's worth our time to argue over it.

Now we come to your statements: "There is not a disagreement. There is not a controversy. It's such a simple question that everyone agrees. They just phrase it differently." I think you may have a point of view that I might learn something from, so I have a few questions I'm interested in.

First, I did a lookup of "disagree" on the computer and it stated that "disagree" means "have or express a different opinion". For example, Oscar Perron and Abraham Fraenkel disagree because one of them states that Cantor's proof for the existence of transcendental numbers is a non-constructive proof while the other states that this proof is constructive. So Perron and Fraenkel have and express a different opinion. However, I do agree that it's not a controversy, which is defined as a "disagreement, typically when prolonged, public, and heated." Mostly one side ignores the other so it never gets heated. Are your definitions of "disagree" or "controversy" different from mine?

Now for your statements that I find a bit cryptic but very interesting: "It's such a simple question that everyone agrees. They just phrase it differently." Please explain why "It's such a simple question". What is the question? Also, why do you think that everyone agrees? Finally, how do they phrase it? Thank you,

@RJGray: What I mean is that the modifications of the argument to produce a specific real number are so straightforward that it is not really plausible that Perron and Fraenkel actually dispute it. Either they haven't seen it (Fraenkel I think died quite a long time ago so that's a bit of a different case), or they consider it to be enough extra argument that it's no longer really "Cantor's argument". Neither of those possibilities is very interesting, and neither constitutes a real disagreement about whether the argument gives an explicit real. The second possibility is, at most, a disagreement about what counts as "Cantor's argument", which is much less substantive than the language made it sound. --Trovatore (talk) 20:03, 5 May 2020 (UTC)

Read: [[User:RJGray/Translate#Reason for ref [2] in Lead]] BEFORE WRITING TO IRY-HOR.

User talk:Iry-Hor/Archive 11#Cantor at FAC

This article is about Cantor's first article on infinite sets, which contains his discovery of two kinds of infinite sets: countable sets and uncountable sets. The members of a countable set, such as the fractions between 0 and 1, can be written as a sequence, for example 1/2, 1/3, 2/3, 1/4, 3/4, … . The members of an uncountable set, such as the real numbers between 0 and 1, cannot be written as a sequence. The significant developments in mathematics that came from the use of countable and uncountable sets justify the importance of this article. Also, it would be good to have another featured article on mathematics: of the 5752 featured articles, only 18 (about 0.3%) are on mathematics.

Number of featured articles:

Wikipedia:Featured articles (4/5/2020: 5,741) (4/19: 5740) (4/24: 5745) (4/28: 5752)

English_Wikipedia#Wikiprojects,_and_assessments_of_articles'_importance_and_quality (4/5/2020: 6,866) [Updated daily by a bot NOT!!]

Cantor's first set theory article Refs in lead

Translate Cantor draft

Wikipedia:Featured_articles#Mathematics Cantor's Absolute Infinite Acta Mathematica

Axiom of limitation of size Von Neumann–Bernays–Gödel set theory Sandbox100 Sandbox101 Sandboxcantor1 Sandboxcantor2

Math TODO:

MOS:MATH Wikipedia:WikiProject Mathematics#Featured content

MOS:ACCESS MOS:TABLECAPTION

Georg Cantor Cantor's diagonal argument Controversy over Cantor's theory Axiom of dependent choice Function (set theory) GA_Review Representation theory of finite groups Lorentz group Axiom schema of replacement set theory ZFC Zermelo set theory

WP:FAC WP:Featured articles WP:Good articles WP:Mentoring for FAC Displaying a formula

Wikipedia:File copyright tags/All

Talk:Georg Cantor's first set theory article/GA2 User_talk:D.Lazard User_talk:RJGray

WP:Verifiability, not truth WP:Good articles

https://en.wikiversity.org/wiki/WikiJournal_of_Science

Dauben, Joseph (1979), Georg Cantor: His Mathematics and Philosophy of the Infinite, Harvard University Press, ISBN 9781503521513 {{citation}}: Invalid |ref=harv (help).

Michael Hardy

Hi Michael, The article you started years ago (under the name "Cantor's first uncountability proof") will soon be on its way to (hopefully) becoming a Featured Article. Iry-Hor is an excellent mentor. He gave me a list of 16 items to fix or consider. I made a number of changes and he now advises me to nominate the article. I will be waiting until January when I will have the time to give quick responses to the reviewers. The latest copy of the article can be found at User:RJGray/Sandbox100.

One place that needed changing was the lead. The current lead does not include the article title in bold, which the MOS requires. Iry-Hor suggested a couple of changes that put the article title at the beginning. Here's the first sentence of the new lead:

Cantor's first set theory article contains Georg Cantor's first theorems of transfinite set theory, which studies infinite sets and their properties.

The article title has been changed to start with "Cantor's" rather than "Georg Cantor's". This has several advantages:

WP:Article titles#Conciseness states: "The goal of conciseness is to balance brevity with sufficient information to identify the topic to a person familiar with the general subject area." It goes on to state "Exceptions exist for biographical articles. For example, neither a given name nor a family name is usually omitted or abbreviated for conciseness." The Cantor article doesn't qualify for this exception. Since I'm trying to meet the standards for a featured article, it's best for me to follow the article title rules.
It conforms to the article titles used for other mathematicians' works. For example, there are article titles beginning with "Zermelo's", "Dedekind's", and "Von Neumann's". Of these, only "Von Neumann's" has a redirect: "John von Neumann's inequality". Since these article titles all begin with "Von Neumann's", a search gives all files beginning with "Von Neumann's", whereas a search on "Cantor's" currently leaves out "Cantor's first set theory article"—it's left out since it's a redirect rather than an article title. I use this type of search to delve deeper into a mathematician's work.
A featured article should have the best possible opening. A repeat of "Georg Cantor" in the first sentence is redundant. Also, without the repeat, the wikilinked "Georg Cantor" is the first occurrence of "Georg Cantor".

By the way, the current article title starts with "Georg Cantor's" because when I rewrote the article, I made the first mistake in WP:MOS#Avoid these common mistakes: "Links should not be placed in the boldface reiteration of the title in the opening sentence of a lead". You corrected that mistake. I'm now an avid reader of the MOS since I'm working on making it a featured article.

I thought it might be a good learning experience for me to handle the page move from "Georg Cantor's first uncountability proof" to "Cantor's first uncountability proof". However, I've already learned that I'll have to make a technical request because the "Cantor's first set theory article" redirect page has two items in its history. Could you make the move for me? My reason for the move is: "Article title change. New title is more concise."

Thank you for all the help and encouragement you've given me since I started writing for Wikipedia. In fact, your Cantor article nominations for GA and DYK last year made me interested in nominating it for FA. —~

Iry-Hor

Wikipedia:Manual of Style/Mathematics Wikipedia:WikiProject Mathematics/Proofs Help:Pictures Cantor's 1874 uncountability proof

Today's Featured Article: Check when a math article last appeared. Go to WP:TFA.

RJGray I think your article is really excellent. You can confidently propose it at FAC now! Write a very short blurb to present it at FAC and go for it. I will support the article and I am sure plenty of other people will do so too. As I said, once this is FA (FAC can take a couple of months so it will be next year), I strongly suggest you propose it at Today's Features Article Candidates.Iry-Hor (talk) 08:34, 5 December 2019 (UTC)

Hi Iry-Hor, That was fast! I didn't realize I was so close to a FA. I was going to ask you for an example blurb, but I've found them myself. For example, I found one at the beginning of Wikipedia:Featured article candidates/Featured log/November 2019#Decipherment of ancient Egyptian scripts. With the holiday season approaching, I'm going to be very busy so I won't have that much time to fix things. I have no idea of what kind of response time they expect on a FAC. Should I wait or go ahead? Of course, I first have to write the blurb and knowing myself, I will read quite a few blurbs first, and then write and rewrite mine quite a few times. Thanks again for the work you have done in making excellent suggestions for the article. -RJGray (talk)

Ferreiros

The Early Development of Set Theory: The Stanford Encyclopedia of Philosophy (Summer 2019 Edition)

Although Cantor might have found that paradox as early as 1883, immediately after introducing the transfinite ordinals (for arguments in favour of this idea see Purkert & Ilgauds 1987 and Tait 2000), the evidence indicates clearly that it was not until 1896/97 that he found this paradoxical argument and realized its implications. By this time, he was also able to employ Cantor’s Theorem to yield the Cantor paradox, or paradox of the alephs: if there existed a “set of all” cardinal numbers (alephs), Cantor’s Theorem applied to it would give a new aleph ℵ, such that ℵ<ℵ. The great set theorist realized perfectly well that these paradoxes were a fatal blow to the “logical” approaches to sets favoured by Frege and Dedekind. Cantor emphasized that his views were “in diametrical opposition” to Dedekind’s, and in particular to his “naïve assumption that all well-defined collections, or systems, are also ‘consistent systems’ ” (see the letter to Hilbert, Nov. 15, 1899, in Purkert & Ilgauds 1987: 154). (Contrary to what has often been claimed, Cantor’s ambiguous definition of set in his paper of 1895 was intended to be “diametrically opposite” to the logicists’ understanding of sets—often called “naïve” set theory, or can more properly be called the dichotomy conception of sets, following a suggestion of Gödel.)

Cantor thought he could solve the problem of the paradoxes by distinguishing between “consistent multiplicities” or sets, and “inconsistent multiplicities”. But, in the absence of explicit criteria for the distinction, this was simply a verbal answer to the problem. Being aware of deficiencies in his new ideas, Cantor never published a last paper he had been preparing, in which he planned to discuss the paradoxes and the problem of well-ordering (we know quite well the contents of this unpublished paper, as Cantor discussed it in correspondence with Dedekind and Hilbert; see the 1899 letters to Dedekind in Cantor 1932, or Ewald 1996: vol. II). Cantor presented an argument that relied on the “Burali-Forti” paradox of the ordinals, and aimed to prove that every set can be well-ordered. This argument was later rediscovered by the British mathematician P.E.B. Jourdain, but it is open to criticism because it works with “inconsistent multiplicities” (Cantor’s term in the above-mentioned letters).

Michael Hardy: Addition to article & Featured article nomination

Hi Michael, I've decided that using Cantor's 1874 uncountability proof is shorter and better than using Cantor's first published uncountability proof. I've also changed Cantor's second uncountability proof to Cantor's 1879 uncountability proof. I plan to change it at all 16 references. Shall I play it safe and keep the

Hi Michael,

I've added another proof to Georg Cantor's first set theory article. The current Wikipedia article contains two of Cantor's three pre-diagonal uncountability proofs. I've added the third proof, which came before the other proofs but is in a letter that wasn't published until 1937. My changes appear in User:RJGray/Sandbox100. I've added the proof to the section: The development of Cantor's ideas. I also modified the paragraphs that precede and follow the proof.

I also had to change the current lead that states: "This theorem is proved using Cantor's first uncountability proof, which differs from the more familiar proof using his diagonal argument." The proof in Cantor's 1874 article is actually Cantor's first published uncountability proof since Cantor's letter contains the first proof but was published later. I now use Cantor's first published uncountability proof. An alternative wording is: Cantor's first proof of uncountability if you think that it's better. I still have to change the references to Cantor's first uncountability proof that occur in 16 other Wikipedia articles.

I revisited the Cantor article because I'm seriously considering nominating it for Featured Article (the new addition makes it more comprehensive). I'm glad you nominated the article for Good Article and DYK. For Featured Article, it's stated that "Nominators must be sufficiently familiar with the subject matter and sources to deal with objections during the featured article candidates (FAC) process.": see WP:FAC. Since I'm most familiar with the sources, I think that I should nominate it.

It's also recommended that I find a mentor before nominating the article. The mentor page (see WP:FAM#Mentors) lists several editors for math or science:

User:Iry-Hor: My Wiki work is exclusively on Ancient Egypt, but I can help in Ancient History, and Mathematics and Physics articles.

User:Ceranthor: Science, biography, mythology.

User:Casliber: Biology, astronomy, other sciencey stuff, sports, popular culture.

User:Cwmhiraeth: I prefer scientific or general topics, and am totally uninterested in popular culture.

I welcome any suggestions you may have on who would be a good mentor, either one of the above or someone else. I found my experiences with the Good Article and DYK nominations to be a lot of work, but it was well worth it. It improved the article and I learned a lot about writing good Wikipedia articles. I look forward to the suggestions I get from you, from a mentor, and from a featured article review.

Thanks, RJGray (talk) 00:54, 15 February 2016 (UTC)

ADD TO: "The development of Cantor's ideas"

(ALSO: CHANGE section "Cantor's second uncountability proof" to "Cantor's 1879 uncountability proof").

Georg Cantor published his first set theory article in 1874, and it contains the first theorems of transfinite set theory, which studies infinite sets and their properties.^[1] One of these theorems is "Cantor's revolutionary discovery" that the set of all real numbers is uncountably, rather than countably, infinite.^[2] This theorem is proved using Cantor's first published proof of uncountability,^[A] which differs from the more familiar proof using his diagonal argument. The title of the article, "On a Property of the Collection of All Real Algebraic Numbers" ("Ueber eine Eigenschaft des Inbegriffes aller reellen algebraischen Zahlen"), refers to its first theorem: the set of real algebraic numbers is countable. In 1879, Cantor modified his uncountability proof by using the topological notion of a set being dense in an interval.

On December 7, Cantor sent Dedekind a proof by contradiction that the set of real numbers is uncountable. Cantor starts by assuming that the real numbers in $[0,1]$ can be written as a sequence. Then, he uses this sequence to produce a number in $[0,1]$ that is not in the sequence, thus contradicting his assumption.^[3]

Cantor's December 7, 1873 uncountability proof
The proof is by contradiction and starts by assuming that the real numbers in $[0,1]$ can be written as a sequence: $(\mathrm {I} )\;\;\omega _{1},\,\omega _{2},\,\omega _{3},\,\ldots ,\,\omega _{n},\,\ldots$ An increasing sequence is extracted from this sequence by letting $\omega _{1}^{1}=$ the first term $,\ \omega _{1}^{2}=$ the next largest term following $\omega _{1}^{1},\ \omega _{1}^{3}=$ the next largest term following $\omega _{1}^{2},$ and so forth. The same procedure is applied to the remaining members of the original sequence to extract another increasing sequence. By continuing this process of extracting sequences, one sees that the sequence $(\mathrm {I} )$ can be decomposed into the infinitely many sequences:^[4] $(1)\;\;\omega _{1}^{1},\,\omega _{1}^{2},\,\omega _{1}^{3},\,\ldots ,\,\omega _{1}^{n},\,\ldots$ $(2)\;\;\omega _{2}^{1},\,\omega _{2}^{2},\,\omega _{2}^{3},\,\ldots ,\,\omega _{2}^{n},\,\ldots$ $(3)\;\;\omega _{3}^{1},\,\omega _{3}^{2},\,\omega _{3}^{3},\,\ldots ,\,\omega _{3}^{n},\,\ldots$ $\ \ \vdots$ Let $[p,q]$ be an interval such that no term of sequence (1) lies in it. For example, let $p$ and $q$ satisfy $\omega _{1}^{1}<p<q<\omega _{1}^{2}.$ Then $\omega _{1}^{1}<p<q<\omega _{1}^{n}$ for $n\geq 2,$ so no term of sequence (1) lies in $[p,q].$ ^[4] Now consider whether the terms of the other sequences lie outside $[p,q].$ All terms of some of these sequences may lie outside of $[p,q];$ however, there must be some sequence such that not all its terms lie outside $[p,q].$ Otherwise, the numbers in $[p,q]$ would not be contained in sequence $(\mathrm {I} ),$ contrary to the initial hypothesis. Let sequence $(k)$ be the first sequence that contains a term in $[p,q]$ and let $\omega _{k}^{n}$ be the first term. Since $p<\omega _{k}^{n}<q,$ let $p_{1}$ and $q_{1}$ satisfy $p<p_{1}<q_{1}<\omega _{k}^{n}<q.$ Then $[p,q]\supset [p_{1},q_{1}]$ and the terms of sequences $(1),(2),\ldots ,(k-1)$ lie outside of $[p_{1},q_{1}].$ ^[4] Repeat the above argument starting with $[p_{1},q_{1}]\!:\,$ Let sequence $(k_{1})$ be the first sequence containing a term in $[p_{1},q_{1}]$ and let $\omega _{k_{1}}^{n}$ be the first term. Since $\ p_{1}<\omega _{k_{1}}^{n}<q_{1},$ let $p_{2}$ and $q_{2}$ satisfy $p_{1}<p_{2}<q_{2}<\omega _{k_{1}}^{n}<q_{1}.$ Then $[p_{1},q_{1}]\supset [p_{2},q_{2}]$ and the terms of sequences $(k_{1}),\ldots ,(k_{2}-1)$ lie outside of $[p_{2},q_{2}].$ ^[4] One sees that it is possible to form an infinite sequence of intervals $[p,q]\supset [p_{1},q_{1}]\supset [p_{2},q_{2}]\supset \ldots$ such that: The members of the $1^{st},2^{nd},\ldots ,(k-1)^{st}$ sequence lie outside $[p,q];$ The members of the $k^{th},\ldots ,(k_{1}-1)^{st}$ sequence lie outside $[p_{1},q_{1}];$ The members of the $(k_{1})^{th},\ldots ,(k_{2}-1)^{st}$ sequence lie outside $[p_{2},q_{2}];$ $\ldots$ ^[4] Cantor's construction of the infinite sequence of intervals is informal. A formal construction would use an inductive proof. For uniform notation, $p_{0},q_{0},$ and $k_{0}$ is used instead of $p,q,$ and $k.$ The base case of the inductive proof is handled the same way as Cantor's construction: it takes an interval $[p_{0},q_{0}]$ such that no term of sequence (1) lies in it. The inductive step uses Cantor's construction applied just once to $[p_{n},q_{n}]$ rather than successively to $[p_{0},q_{0}],[p_{1},q_{1}],[p_{2},q_{2}],\ldots .$ Since $p_{n}$ and $q_{n}$ are bounded monotonic sequences, the limits $\lim _{n\to \infty }p_{n}$ and $\lim _{n\to \infty }q_{n}$ exist. Also, $p_{n}<q_{n}$ for all $n$ implies $\lim _{n\to \infty }p_{n}\leq \lim _{n\to \infty }q_{n}.$ Hence, there is at least one number $\eta$ in $(0,1)$ that lies in all the intervals $[p,q]$ and $[p_{n},q_{n}].$ Namely, $\eta$ can be any number in $[\lim _{n\to \infty }p_{n},\lim _{n\to \infty }q_{n}].$ This implies that $\eta$ lies outside all the sequences $(1),(2),(3),\ldots ,$ contradicting the initial hypothesis that sequence $(\mathrm {I} )$ contains all the real numbers in $[0,1].$ Therefore, the set of all real numbers is uncountable.^[4]

Dedekind received Cantor's proof on December 8. On that same day, Dedekind simplified the proof and sent his proof to Cantor. Cantor used Dedekind's proof in his article.^[5] The letter containing Cantor's proof was not published until 1937.^[6]

On December 9, Cantor stated the theorem that allowed him to construct transcendental numbers as well as prove the uncountability of the set of real numbers:

I show directly that if I start with a sequence
$(\mathrm {I} )\;\;\omega _{1},\,\omega _{2},\,\omega _{3},\,\ldots ,\,\omega _{n},\,\ldots ,$

then in every given interval $[\alpha ,\beta ],$ I can determine a number $\eta$ that is not included in $(\mathrm {I} )$ .^[7]

(*** SEE Current article for rest of section. ***)

Notes

^ This proof is a simplification of an earlier proof that Cantor that had sent to Richard Dedekind.

References

^ Ferreirós 2007, p. 171.
^ Dauben 1993, p. 4.
^ Noether & Cavaillès 1937, pp. 14–15. English translation: Ewald 1996, pp. 845–846.
^ ^a ^b ^c ^d ^e ^f Cite error: The named reference Dec7letter was invoked but never defined (see the help page).
^ Noether & Cavaillès 1937, p. 19. English translation: Ewald 1996, p. 849.
^ Ewald 1996, p. 843.
^ Noether & Cavaillès 1937, p. 16. English translation: Gray 1994, p. 827.

Stuff for monthly article

Georg Cantor published three uncountability proofs. His first proof was discovered in a letter he wrote to Richard Dedekind and was published in 19__ by Emmy Noether and _____________ __________. Cantor's use of a normal form and how it led him to discover a proof.

Dedekind's simplifications

Removed need for normal form.
1. Removed Cantor's use of $(k_{n})$ sequences by using subinterval $[\omega _{n}^{1},\omega _{n}^{2}]$ for $[p_{n},q_{n}].$
2. Saw that instead of using a subinterval of $[\omega _{n}^{1},\omega _{n}^{2}]$ could use next two terms within $[p_{n},q_{n}].$
Saw that the above changes can produce a real that does not belong to a given arbitrary sequence of reals, so there is need for assumption that positive reals less than one can be written as a sequence.

Other to do

Dedekind's contributions to Cantor's article: More details on Cantor's use of Dedekind's ideas. Especially, Cantor handling of Dedekind's proof. Also, more details on the relationship between Cantor & Dedekind and more about Cantor's quick reply to Dedekind--how Dedekind first thought Cantor didn't get the letter and later he had second thoughts after Cantor printed word-for-word two of Dedekind's proofs that appear in their correspondence (??). Also, the price Cantor and Dedekind (and others?) paid.

^[1]

Notes

References

^ Kanamori 2009, pp. 48, 58. Bernays' articles are reprinted in Müller 1976, pp. 1–117.

Jochen (#1: Reference within efn)

Hi Jochen. I was wondering if you could help me enlarge a photo. In fact, it's the photo of Paul Bernays that you extracted.

Reference within efn: Chaplin believed he was born in South London.^[a]

Reference within efn: Chaplin believed he was born in South London.^[b]

Reference within efn: Chaplin believed he was born in South London.^[c] ^[5] End efn #3}}

Notes

^ Begin efn #1 ^[1] End efn #1
^ Begin efn #2 ^[3] End efn #2
^ Begin efn #3 ^[4]

References

^ Outer reference (template): record of Chaplin's birth. Date of his birth. End of outer reference
^ Inner ref: Robinson, p. xxiv. End of inner ref
^ Outer reference (template): record of Chaplin's birth. Date of his birth.^[2] End of outer reference
^ Inner ref: Robinson, p. xxiv. End of inner ref
^ Inner ref #3

Need some advice

I am contacting you because of your comments in Wikipedia:Verifiability/2012 RfC:

Strong support but move the note into the text. Rationale for moving the note: firstly, notes should be avoided in a lead. Secondly "verifiability, not truth" is a sufficiently longstanding and notable slogan for deserving to be clearly apparent in the lead. Otherwise this version is clear, simple and is the one which best resolves the following ambiguity. In a scientific context, especially in mathematics, "verifiability" and "truth" have not the same meaning as in Wikipedia: A theorem is true if and only if it has been proved, and a proof is correct if and only if it is verifiable by anybody. This ambiguity makes, sometimes, content discussions very difficult with unexperienced editors and also with some experienced editors with insufficient mathematical knowledge. A policy is aimed to make discussions easier, not harder. This version is the one that best resolves this ambiguity. D.Lazard (talk) 12:29, 3 July 2012 (UTC)

Recently, an article I rewrote a few years ago, Georg Cantor's first set theory article, was nominated for Good Article (see Talk:Georg Cantor's first set theory article/GA2). With feedback from the GA reviewer, I made improvements and the article achieved GA status. I did have some minor difficulties over my use of proofs, but I made an improvement and successfully cited WP:Scientific citation guidelines#Examples, derivations and restatements.

Next, the article was nominated for "Did you know?". The DYK editor has problems with at least two simple proofs that Cantor failed to give because he was communicating with his fellow research mathematicians. Since Cantor did not give these proofs, I provided them. I know that Wikipedia appeals to a wide audience, I wanted to make sure readers had a complete proof rather than expecting them to finish it. Currently, we are dealing with:

Proof of Cantor's uncountability theorem: I state that "Cantor does not explicitly prove his uncountability theorem, which follows easily from his second theorem." Then I supply a simple derivation of his uncountability theorem, which uses his second theorem.
The development of Cantor's ideas: I state "2. The proof by contradiction used to prove the existence of transcendental numbers from the countability of the real algebraic numbers and the uncountability of real numbers. Cantor's December 2nd letter mentions this existence proof but does not contain it." Then I supply a simple derivation (or proof) of the existence of transcendental numbers given the countability of the real algebraic numbers and the uncountability of real numbers.

I justify my derivations or proofs by the scientific citation guidelines, which not only permits the use of "simple derivations" but even allows an editor a further option: "it is often necessary … to provide a different derivation". I guess it could be said that I may be using a different derivation since Cantor gave no derivation.

I am currently involved in a discussion with this editor and would appreciate any advice you have. The DYK discussion is at Talk:Georg Cantor's first set theory article#Did you know nomination.

By the way, how did that RfC go? Also, I just checked WP:Good Article and found that there are only 63 good articles in mathematics. This is by far the least of any subject. Could be this caused at least partly by what you bring up in the RfC? Namely, "This ambiguity makes, sometimes, content discussions very difficult with unexperienced editors and also with some experienced editors with insufficient mathematical knowledge."

My plan is not to give up on GAs and DYKs. I think that mathematics needs more GAs. I didn't do the current GA nomination—the editor who wrote the first version of the article did. However, next summer I plan to nominate an article that I recently rewrote. Thank you, RJGray (talk) 17:44, 23 September 2018 (UTC)

I am not really interested in GAs and DYKs. My point is rather the number of articles that are among the most viewed ones and are (or should be) rated C. See my user page for having examples of what I mean. Generally I push such articles toward level B or A, but not higher as it is difficult for me to find sources that I consider as belonging to the common knowledge. Nevertheless, I try to add only sourcable content. When I know that sources are difficult to find for some reasons, I use WP:CALC (after all, a source is a computation), and provide a proof for insuring verifiability (for an example, see my edits on the relations between monomorphisms and epimorphisms one a one side, and injections and surjections on the other side, in Homomorphism: as this concerns many kinds of homomorphisms, it is difficult to find a source covering all of them).

On the other hand, I have often difficulties with articles on mathematical logic and their editors: As they like formal treatment, the intuitive interpretation is often laking, even if it has motivated the inventors of the theory. When I try to add some intuitive explanation, I am generally reverted, as an intuitive explanation is always formally wrong.

Otherwise, I am unable to give general advices. If you have specific point of discussion, I am ready to give my advice on them, but I need a specific link. D.Lazard (talk) 21:00, 23 September 2018 (UTC)

Reply

Thank you for your patience. Even though I haven't changed my opinion that sources are not needed in this case, I decided that finding sources would be a good challenge of my knowledge of the literature. It took a bit of work but I suceeded. I have updated the Wikipedia article.

The source I found for my proof of the existence of transcendental numbers is Perron's book, which is in German and has not been translated. However, in my research on Wikipedia policies, I learned how to handle this: I've put the German proof and my translation of it in a note (see WP:V#Quoting). The source I found for my proof of Cantor's uncountability theorem is the article "Georg Cantor and Transcendental Numbers".

I must say that I am happier having the sources than not having them—I like to saturate Wikipedia articles with citations. I regard citations as doorways to deeper knowledge of a subject.

However, I think that Wikipedia's verifiability policy of citations and reliable sources can be characterized as "passive verifiability". By this I mean that the citations mostly sit there passively with few people actively looking many of them up. Consider my article with its 63 references with one reference containing 6 citations. How many citations will the typical reader of an article look up? Probably far fewer than 63. On the positive side, the citations are available if issues come up.

Now consider an article's mathematical proofs. Proofs actively engage readers who think through them and decide if they are correct or not. And if they aren't correct, they may modify them. So proofs are an example of "active verifiability". I always read mathematical proofs carefully and when I find one that is inaccurate or not clear, I rewrite it. So the accuracy of a mathematical proof depends not just on the person who initially writes it, but on everyone who reads it, thinks about it, and improves it.

On the other hand, I seldom look up the citations of a Wikipedia article unless I'm interested in reading further on the subject. I know it's hard work checking sources—the last thing I do before posting an article I wrote is to check all my citations for accuracy.

Another weakness of Wikipedia's verifiability policy is that reliable sources may have errors. The section "The disagreement about Cantor's existence proof" gives examples. In fact, the books asserting that Cantor's existence proof is non-constructive outnumber the books asserting asserting his proof is constructive.

I wish to thank you for suggesting your improvement. This article has now benefited from two GA reviews (it failed the first one but that motivated me to do a second rewrite of the article) and one DYK review. —RJGray (talk) 19:20, 25 September 2018 (UTC)

This issue is important for me to because I suspect it will occur in the future. I tend to work in history of mathematics and may be faced again with the fact that research mathematicians tend to leave out simple derivations or proofs when communicating with their fellow research mathematicians via articles and letters. The only reason that I provided my own simple derivations was because Cantor left out two: one in his article and the other in a letter to Dedekind. I could follow Cantor and skip the derivations. However, because Wikipedia appeals to a wide audience, I wanted to make sure readers had a complete proof down to small details.

I need to understand exactly what in my rewrite you find to be a problem. I think it is only two items:

Proof of Cantor's uncountability theorem: I state that "Cantor does not explicitly prove his uncountability theorem, which follows easily from his second theorem." Then I supply a simple derivation of his uncountability theorem, which uses his second theorem.
The development of Cantor's ideas: I state "2. The proof by contradiction used to prove the existence of transcendental numbers from the countability of the real algebraic numbers and the uncountability of real numbers. Cantor's December 2nd letter mentions this existence proof but does not contain it." Then I supply a simple derivation (or proof) of the existence of transcendental numbers given the countability of the real algebraic numbers and the uncountability of real numbers.

I justify my use by the scientific citation guidelines, which not only permits the use of "simple derivations" but even allows an editor a further option: "it is often necessary … to provide a different derivation". I guess it could be said that I may be using a different derivation since Cantor gave no derivation.

So it appears to me that the scientific citation guidelines may be carving out a small exception to WP:V similar to the translation exception in WP:V#Quoting occurs in WP:RS. Please note that in the last two sentences it addresses verifiability and acknowledges that the verifiability dependence is not just on the translating editor, but on all editors who know the original language and the language being translated to. This is similar to two math derivations, whose verifiability depends on all editors who know the math it is built on.

If you quote a non-English reliable source (whether in the main text or in a footnote), a translation into English should always accompany the quote. Translations published by reliable sources are preferred over translations by Wikipedians, but translations by Wikipedians are preferred over machine translations. When using a machine translation of source material, editors should be reasonably certain that the translation is accurate and the source is appropriate.

Here we do not depend on just the editor who did the translation, but all also editors who know the original language and the language that it is being translated to. In this way, we get some verifiability.

As for mathematics, it has a stronger form of verifiability than Wikipedia uses—namely, mathematical proof, which has a very precise definition that allow one to prove theorems that will always be valid. Since a mathematical proof has a precise definition, it can be verified by anyone knowing the math involved.

In fact, the history of this Cantor article is an example of how mathematical proof is a stronger form of verifiability than Wikipedia uses. The original article begins: "Georg Cantor's first uncountability proof demonstrates that the set of all real numbers is uncountable. Cantor formulated the proof in December 1873 and published it in 1874 in Crelle's Journal[1] …" The article went on to give the proof by contradiction that Cantor never published. Being familiar with an article that documents this error, I decided to rewrite the Wikipedia article despite the fact that I had never before contributed to Wikipedia. Michael Hardy, the article's original editor, was extremely helpful. In fact, Michael's enthusiasm for my work led me to become a regular contributor to Wikipedia.

I find it interesting that the reason I rewrote this article was because it had a serious error despite satisfying Wikipedia's verifiability criterion. The original article, titled "Cantor's first uncountability proof", cited reliable sources that claimed Cantor's proof was a pure existence proof that did not construct transcendental numbers. I used the source Cantor and Transcendental Numbers. In his review of this source, Ivor Grattan-Guinness, a historian of mathematics who has written extensively on set theory, stated: "It is commonly believed that Cantor's proof of the existence of transcendental numbers, published in 1874, merely proves an existence theorem. The author refutes this view by using a computer program to determine such a number."

So the current Wikipedia article is correcting a common error in the math literature.

I think that clarification is needed in the scientific citation guidelines section if it is in conflict with WP:V such as in the case of translation that is mentioned in WP:RS. Also it would be nice if WP:V or WP:RS used this example as a case where "reliable sources" is weaker than the standards of a particular domain (in this case, mathematics). I know that WP:RS#Age matters deals with this to some extent, but in the current case, the latest source that states Cantor's proof is non-constructive is a few years newer than the latest source that states his proof is constructive.

It will take me a several days to write something for Wikiproject Mathematics. I will be quoting from your most recent post and I will first post what I write on this page so you can make sure that I accurately capture what your concerns are.

Notes

References

Havil irrational book

However, I found a book that has this example except that it uses the enumeration 1/2, 1/3, 1/4, 2/4, 3/4, ... while I omit reducible fractions. Interestingly, the book is dated 2012 and this example has been in Wikipedia since Feb. 19, 2010. However, the book may have got the idea from Gray 1994, which gives this example in an exercise for the reader (see p. 283). So I can add this book or the Gray 1994 article as a reference. However, the 1994 article doesn't say that the number generated is sqrt(2) - 1. (If I do add the book as a reference, should I also add a note that Wikipedia gave this same example earlier? This would prevent any claim of copyright violation.) The book is: Havil, Julian (2012), The Irrationals: A Story of the Numbers You Can't Count On, Princeton University Press, ISBN 978-0-691-14342-2 {{citation}}: Invalid |ref=harv (help). The example is on pp. 208—209: see The Irrationals, p. 208.

Dauben uses examples from Cantor's article to show Kronecker's influence.^[1]

(2^{\aleph _{0}})^{n}=2^{\aleph _{0}\times n}=2^{\aleph _{0}}.

Add to Infinity??

Cardinal arithmetic can be used to show not only that the number of points in a real number line is equal to the number of points in any segment of that line (can be shown with projection) (a, b) -> (-inf, +inf) but that this is equal to the number of points on a plane and, indeed, in any finite-dimensional space.^[2]

In set theory, the definition of a function is simply a set or class of ordered pairs $(x,y)$ such that every $x$ is associated with only one $y$ . That is, F is a function if F is a set or class of ordered pairs such that for all x, y, and z: If (x, y) ∈ F and (x, z) ∈ F, then y = z.

of a function must be a set. This is usually not a problem since, in general, it is not difficult to consider only functions whose domain and codomain are sets even if the domain is not explicitly defined.

Let $f\colon X\to Y.$ The image by $f$ of an element $x$ of the domain $X$ is $f (x)$ . If $A$ is any subset of $X$ , then the image of $A$ by $f$ , denoted $f (A)$ is the subset of the codomain $Y$ consisting of all images of elements of $A$ , that is,

f(A)=\{f(x)\mid x\in A\}.

The image of $f$ is the image of the whole domain, that is $f (X)$ . It is also called the range of $f$ , although the term may also refer to the codomain.^[3]

However, set theory requires a more general definition of both function and image. In set theory, the definition of function does not mention a domain or a codomain, and the definition of the image F[A] does not require that A be a subset of the domain.^[4] Set theory defines the domain of $F$ as: $Dom(F)=\{x:\exists y[(x,y)\in F]\}.$ The range of $F$ is defined as: $Rng(F)=\{y:\exists x[(x,y)\in F]\}.$

The definition of a function that is given in this article requires the concept of set, since the domain and the codomain of a function must be a set. This is usually not a problem since, in general, it is not difficult to consider only functions whose domain and codomain are sets even if the domain is not explicitly defined.

However, set theory requires a more general definition of both function and image. In set theory, the definition of function does not mention a domain or a codomain, and the definition of the image F[A] does not require that A be a subset of the domain.^[5] Set theory defines the domain of $F$ as: $Dom(F)=\{x:\exists y[(x,y)\in F]\}.$ The range of $F$ is defined as: $Rng(F)=\{y:\exists x[(x,y)\in F]\}.$

Set theory specializes its function definition to the cases where the domain or both the domain and codomain are specified. This produces the following three definitions.^[6]

F is a function if F is a set or class of ordered pairs such that for all x, y, and z: If (x, y) ∈ F and (x, z) ∈ F, then y = z.
F is a function on X if F is a function and X is the domain of F.
F is a function from X to Y if F is a function, X is the domain of F, and F[X] ⊆ Y.

Axiom of replacement. If $F$ is a function and $a$ is a set, then $F[a]$ , the image of $a$ under $F$ , is a set.

\forall F\,\forall a\,[F{\text{ is a function}}\implies \exists b\,\forall y\,(y\in b\iff \exists x(x\in a\,\land \,(x,y)\in F))].

Not having the requirement $A\subseteq Dom(F)$ in the definition of $F[A]$ produces a stronger axiom of replacement, which is used in the following proof.

Theorem (NBG's axiom of separation). If $A$ is a set and $B$ is a subclass of $A,$ then $B$ is a set.
Proof: The class existence theorem constructs the restriction of the identity function to $B$ : $I{\upharpoonright _{B}}=\{(x_{1},x_{2}):x_{1}\in B\land x_{2}=x_{1}\}.$ Since the image of $A$ under $I{\upharpoonright _{B}}$ is $B$ , the axiom of replacement implies that $B$ is a set. This proof depends on the definition of image not having the requirement $A\subseteq Dom(F)$ since $Dom(I{\upharpoonright _{B}})=B\subseteq A$ rather than $A\subseteq Dom(I{\upharpoonright _{B}}).$

The first definition allows the use of functions without specifying their domain or codomain. For example, this definition is used to state the axiom of replacement for Von Neumann–Bernays–Gödel set theory and ZFC. replacement axiom of von Neumann–Bernays–Gödel set theory uses classes that are functions: For all classes F and for all sets X, if F is a function, then F[X] is a set. Since this definition uses set theory's definition of image, X does not have to be a subset of the domain of F. ZFC's axiom of replacement replaces the class F with a formula that defines F.

The second function definition is used to define a set or class of functions without specifying a codomain. For example, the class of functions on the ordinal α. Also, an infinite sequence can be defined as a function on ω, the set of finite ordinals.

The third definition requires that the function's domain and codomain be specified. For example, the aleph function $\aleph$ is a function from the class of ordinals to the class of infinite cardinals. Its function values are denoted by using a subscript: $\aleph _{\alpha }$ , rather than by using parentheses: $\aleph (\alpha )$ . The notation $\aleph _{\alpha }$ extends the sequence notation $x_{n}$ into the transfinite.

Fixed broken links and rewrote "In set theory"

In general, I think well of the rewrite of this article—it's been long overdue. However, I have three problems with the section "In foundations of mathematics and set theory." Two links to the section—Function (set theory) and Image (set theory)—were broken. These links are used in Von Neumann–Bernays–Gödel set theory (I have fixed them). Also, "foundations of mathematics" is very general and many parts of it do not use functions, such as the parts dealing with various logics. The parts that do use functions, such as model theory, are based on set theory. So I have left "foundations of mathematics" out of the title.

The third problem is that examples were kept in most of the other sections of the "Function" article, but the examples I put into the old "Definitions in set theory" were removed. I think that set theory is as deserving of relevant examples as the other parts of mathematics. I chose examples that are actually used in set theory. The example $x\mapsto \{x\}$ is an artificial example and calling it a "singleton set" is inaccurate—it is a function that maps set $x$ to the singleton set that contains $x.$ My example that is of the same type (a function from a class to a class) is $\aleph _{\alpha }$ . This example also shows how set theory extends the sequence notation to show readers how set theory can extend a common concept of mathematics into the transfinite.

Also, the statement in the current section: "This theory includes the replacement axiom, which may be interpreted as 'if $X$ is a set, and $F$ is a function, then $F [X]$ is a set'." is inaccurate with "may be interpreted". This is the statement of the axiom, not just an interpretation of it. Finally, I wish to point out that this section will soon be linked to from set theory articles other than "Von Neumann–Bernays–Gödel set theory", so it's important to support these articles with examples relevant to set theory.

Overall, I like the new Function article much better than the old one. --RJGray (talk) 21:37, 21 April 2018 (UTC)

History (From "Set Theory")

Mathematical topics typically emerge and evolve through interactions among many researchers. Set theory, however, was founded by a single paper in 1874 by Georg Cantor: "On a Property of the Collection of All Real Algebraic Numbers".^[7]^[8]

Since the 5th century BC, beginning with Greek mathematician Zeno of Elea in the West and early Indian mathematicians in the East, mathematicians had struggled with the concept of infinity. Especially notable is the work of Bernard Bolzano in the first half of the 19th century.^[9] Modern understanding of infinity began in 1870–1874 and was motivated by Cantor's work in real analysis.^[10] An 1872 meeting between Cantor and Richard Dedekind influenced Cantor's thinking and culminated in Cantor's 1874 paper.

Cantor's work initially polarized the mathematicians of his day. While Karl Weierstrass and Dedekind supported Cantor, Leopold Kronecker, now seen as a founder of mathematical constructivism, did not. Cantorian set theory eventually became widespread, due to the utility of Cantorian concepts, such as one-to-one correspondence among sets, his proof that there are more real numbers than integers, and the "infinity of infinities" ("Cantor's paradise") resulting from the power set operation. This utility of set theory led to the article "Mengenlehre" contributed in 1898 by Arthur Schoenflies to Klein's encyclopedia.

The next wave of excitement in set theory started in 1903 when it became generally known that defining a set by specifying its members can lead to contradictions that generate paradoxes. For example, Bertrand Russell and Ernst Zermelo independently found the simplest and best known paradox, now called Russell's paradox: "the set of all sets that are not members of themselves".^[11] This leads to a contradiction since this set must be a member of itself and not a member of itself. In 1903, Russell published his paradox in The Principles of Mathematics.

In an 1896 letter to David Hilbert (CHECK Hilbert letter--did it handle ORD or CARD?) and an 1899 letter to Dedekind, Cantor proved that assuming the multiplicity of all aleph numbers is a set produces a contradiction. Cantor pointed out that this implies that this multiplicity is an inconsistent multiplicity, which cannot be a member of any multiplicity—this prevents the contradiction from creating a paradox.^[12] To see how inconsistent multiplicities prevent paradoxes, consider Russell's paradox: Assume that the multiplicity containing all sets that are not members of themselves is a set. This produces the same contradiction that Russell obtained. Where Russell had an inconsistency because he only had sets, Cantor has only proved that the multiplicity is not a set. Since it is not a set, it cannot be a member of itself. So Russell's paradox, which depends on a set simultaneously being a member of itself and not a member of itself, vanishes. Cantor's letters were not published until 1930 and did not affect the development of set theory. NEED ref; also when was letter to Dedekind published?? Hallett, Michael (1986), Cantorian Set Theory and Limitation of Size, New York: Oxford University Press, pp. 166–167, ISBN 0-19-853283-0.</ref>

The momentum of set theory was such that debate on the paradoxes did not lead to its abandonment. Also, in 1908, Zermelo published his axiom system that has axioms asserting the existence of basic sets—the empty set and an infinite set—and axioms that construct sets from sets whose existence has already been proved. To eliminate the known paradoxes, Zermelo's axiom of separation restricts the specification of members of a set to an existing set. This produces a set that is a subset of the existing set, which eliminates the paradoxes.

In 1930, Zermelo made some modifications to his 1908 axioms and added the axiom of replacement (suggested by Abraham Fraenkel and Thoralf Skolem in 1922) and the axiom of regularity (suggested by John von Neumann in 1925).^[13] The resulting axiom system evolved into ZFC, which became the most commonly used axiom system for set theory.

In 1925, John von Neumann developed an axiom system for sets and classes that eliminates the paradoxes by using an approach similar to Cantor's. A class only has sets as members and can be defined by specifying the sets that are in it. Some classes are sets, but others are not. For example, the class of all sets that are not members of themselves is not a set because the assumption that it is a set produces a contradiction. Since it is not a set, it cannot be a member of itself, so Russell's paradox is avoided. Further work by von Neumann, together with the work of Paul Bernays and Kurt Gödel, led to the von Neumann–Bernays–Gödel (NBG) axiom system.^[14] In the 1960s, it was proved that NBG is a conservative extension of ZFC, which means that both axiom systems prove the same theorems about sets.^[15]

The work of analysts such as Henri Lebesgue demonstrated the great mathematical utility of set theory, which has since become woven into the fabric of modern mathematics. Set theory is commonly used as a foundational system, although in some areas—such as algebraic geometry and algebraic topology-category theory is thought to be a preferred foundation.

Subcompact cardinal (lead)

subcompact cardinal

In mathematics, a subcompact cardinal is a certain kind of large cardinal.

The definition of a subcompact cardinal uses H_κ, which consists of all sets having hereditary cardinality less than κ—that is, sets whose transitive closure only contains sets of cardinality less than κ.

A cardinal κ is subcompact if and only if for every B ⊆ κ⁺, there exists μ < κ, A ⊆ μ⁺, and an elementary embedding j : (H_μ⁺, ϵ, A) → (H_κ⁺, ϵ, B) with critical point μ and j (μ) = κ.

A subcompact cardinal κ is the image of the critical point. In the definition of a quasicompact cardinal, κ is the critical point. A cardinal κ is quasicompact if and only if for every A ⊆ κ⁺, there exists λ > κ, B ⊆ λ⁺, and an elementary embedding j : (H_κ⁺, ϵ, A) → (H_λ⁺, ϵ, B) with critical point κ and j (κ) = λ.

A cardinal number κ is subcompact if and only if for every A ⊂ H(κ⁺) there is a non-trivial elementary embedding j:(H(μ⁺), B) → (H(κ⁺), A).

Analogously, κ is a quasicompact cardinal if and only if for every A ⊂ H(κ⁺) there is a non-trivial elementary embedding j:(H(κ⁺), A) → (H(μ⁺), B) with critical point κ and j(κ) = μ.

Every quasicompact cardinal is subcompact. Quasicompactness is a strengthening of subcompactness in that it projects large cardinal properties upwards. The relationship is analogous to that of extendible versus supercompact cardinals. Quasicompactness may be viewed as a strengthened or "boldface" version of 1-extendibility. Existence of subcompact cardinals implies existence of many 1-extendible cardinals, and hence many superstrong cardinals. Existence of a 2^κ-supercompact cardinal κ implies existence of many quasicompact cardinals.

Subcompact cardinals are noteworthy as the least large cardinals implying a failure of the square principle. If κ is subcompact, then the square principle fails at κ. Canonical inner models at the level of subcompact cardinals satisfy the square principle at all but subcompact cardinals. (Existence of such models has not yet been proved, but in any case the square principle can be forced for weaker cardinals.)

Quasicompactness is one of the strongest large cardinal properties that can be witnessed by current inner models that do not use long extenders. For current inner models, the elementary embeddings included are determined by their effect on P(κ) (as computed at the stage the embedding is included), where κ is the critical point. This prevents them from witnessing even a κ⁺ strongly compact cardinal κ.

Subcompact and quasicompact cardinals were defined by Ronald Jensen.

References

"Square in Core Models" in the September 2001 issue of the Bulletin of Symbolic Logic

Definitions missing a quantifier on B. Changed to use defs from a Schimmerling article; old defs are from Revolvy article "Subcompact cardinal" (I don't know if Revolvy is a reliable source). Check def in Revolvy ref: Schimmerling 2001 article.

Equivalent statements

${\mathsf {DC}}$ is equivalent (over Zermelo–Fraenkel set theory ${\mathsf {ZF}}$ ) to the statement that every pruned tree with $\omega$ levels has a branch.

Proof that $\,{\mathsf {DC}}\iff$ Every pruned tree with ω levels has a branch
$(\,\Longleftarrow \,)$ Let $R$ be an entire binary relation on $X$ . The strategy is to define a tree $T$ on $X$ of finite sequences whose neighboring elements satisfy $R.$ Then a branch through $T$ is an infinite sequence whose neighboring elements satisfy $R.$ Start by defining $\langle x_{0},\dots ,x_{n}\rangle \in T$ if $x_{k}R\,x_{k+1}$ for $0\leq k<n.$ Since $R$ is entire, $T$ is a pruned tree with $\omega$ levels. Thus, $T$ has a branch $\langle x_{0},\dots ,x_{n},\dots \rangle .$ So, for all $n\geq 0\!:$ $\langle x_{0},\dots ,x_{n},x_{n+1}\rangle \in T,$ which implies $x_{n}R\,x_{n+1}.$ Therefore, ${\mathsf {DC}}$ is true. $(\,\Longrightarrow \,)$ Let $T$ be a pruned tree on $X$ with $\omega$ levels. The strategy is to define a binary relation $R$ on $T$ so that ${\mathsf {DC}}$ produces a sequence $t_{n}=\langle x_{0},\dots ,x_{f(n)}\rangle$ where $t_{n}R\,t_{n+1}$ and $f(n)$ is a strictly increasing function. Then the infinite sequence $\langle x_{0},\dots ,x_{k},\dots \rangle$ is a branch. (This proof only needs to prove this for $f(n)=m+n.$ ) Start by defining $u\,R\,v$ if $u$ is an initial subsequence of $v,\operatorname {length} (u)>0,\,$ and $\operatorname {length} (v)=$ $\operatorname {length} (u)+1.$ Since $T$ is a pruned tree with $\omega$ levels, $R$ is entire. Therefore, ${\mathsf {DC}}$ implies that there is an infinite sequence $t_{n}$ such that $t_{n}\,R\,t_{n+1}.$ Now $t_{0}=\langle x_{0},\dots ,x_{m}\rangle$ for some $m\geq 0.$ Let $x_{m+n}$ be the last element of $t_{n}.$ Then $t_{n}=\langle x_{0},\dots ,x_{m},\dots ,x_{m+n}\rangle .$ For all $k\geq 0,$ the sequence $\langle x_{0},\dots ,x_{k}\rangle$ belongs to $T$ because it is an initial subsequence of $t_{0}\,(k\leq m)$ or it is a $t_{n}\,(k\geq m).$ Therefore, $\langle x_{0},\dots ,x_{k},\dots \rangle$ is a branch.

${\mathsf {DC}}$ is also equivalent over ${\mathsf {ZF}}$ to the Baire category theorem for complete metric spaces.^[16]

It is also equivalent over ${\mathsf {ZF}}$ to the Löwenheim–Skolem theorem.^[17]

${\mathsf {DC}}$ is also equivalent over ${\mathsf {ZF}}$ to:

The Baire category theorem for complete metric spaces.
The Baire category theorem for Ċech-complete spaces.

^[18]

Jochen

Gray, Robert (1994), "Georg Cantor and Transcendental Numbers" (PDF), American Mathematical Monthly, 101: 819–832, doi:10.2307/2975129, JSTOR 2975129, MR 1300488, Zbl 0827.01004 {{citation}}: Invalid |ref=harv (help).

Hi Jochen, Thanks for looking over my proposed change. I've updated the article. Concerning du Bois-Reymond and his diagonal argument: It's mentioned on the Talk page: du_Bois-Raymond and Cantor's diagonal argument. Of particular interest is note 1 page 187 in Simmons, Keith (1993). Universality and the Liar: An Essay on Truth and the Diagonal Argument. {{cite book}}: Cite has empty unknown parameter: |1= (help) This note claims that Bois-Reymond's diagonal argument is a different type than Cantor's.

I agree with the Talk page that something should be mentioned, but some research is needed beforehand. Ultimately, it would be nice to have a section covering both sides of this issue.

Here's some interesting material from Stackexchange (of course, we can't use this in a Wikipedia article, but it does say that the diagonal argument is in a footnote on page 365):

He [du Bois-Reymond] refers to [an] 1873 paper where only a particular case was considered. This time he gives a more general version, which Borel highly praised and termed du Bois-Reymond's theorem. A consequence of it, and the motivation, is the non-existence of the "ideal boundary" that can be specified by a sequence of converging/diverging series, as Bertrand earlier proposed (he was thinking of reciprocals to products of powers and powers of iterated logarithms as terms). The du Bois-Reymond's theorem would be the "diagonal argument":

"...if an unlimited family of more and more slowly increasing functions λ1(x), λ2(x), λ3(x) ... is given which for each r satisfies the condition lim λr(X)/λr+1(X) = ∞, one can always specify a function ψ(x) which becomes infinite with x, but more slowly than any function of that family".

The construction of ψ(x) is in a footnote on p. 365, and does show some "diagonality" if one looks hard. There is however no cardinality involved in du Bois-Reymond's setting (Hausdorff will relate gaps to cardinalities only later), so making it into the diagonal argument takes some reading in. In his 1882 book Die allgemeine Functionentheorie (The General Theory of Functions) du Bois-Reymond touches base with Cantorian set theory, and mentions that Cantor showed "continuum of the idealists" to be uncountable. He does not however point out any affinity between the diagonal argument by which it was shown and his earlier construction, let alone lay any claim to it. So whatever the relation between the two it was not apparent to du Bois-Reymond.

NOTE: There is an error in the next to last sentence: du Bois-Reymond book is dated 1882 and Cantor didn't give his diagonal argument until 1891, so du Bois-Reymond couldn't compare his argument with Cantor's diagonal argument. I came across this book years ago and du Bois-Reymond just gives Cantor's 1874 argument. Of course, it can still be argued that if du Bois-Reymond did have a diagonal argument similar to Cantor's later argument, then du Bois-Reymond could have used it to give a new proof of the uncountability of the reals rather than repeating Cantor's 1874 argument.

Rank-into-rank cardinals

Foundations of Mathematics edited by Andrés Eduardo Caicedo, James Cummings, Peter Koellner, Paul B. Larson Beyond I0 by Cramer: pp. 255-256. Rewrite rank-into-rank article to use Kanamori's information.

Gödel/Cohen

Gödel's achievement together with the details of his presentation led to the prominence that NBG would enjoy for the next two decades.^[19] For example, in 1963, Paul Cohen's independence proofs for ZF used tools that Gödel developed for his work in NBG.^[a] Later, ZFC became more popular than NBG. This was caused by several factors, including the extra work required to handle forcing in NBG,^[b] Cohen's 1966 presentation of forcing, which used ZF,^[20] and the proof that NBG is a conservative extension of ZFC.^[c]

^ In his relative consistency proof, Gödel defined a function $F$ on the ordinals such that $F(\alpha )$ is a set built by applying a set operation to sets $F(\beta )$ where $\beta <\alpha .$ The constructible universe is the image of $F.$ Gödel's set operations come from the axiom of pairing, the class existence axioms restricted to sets, and taking the union of the preceding classes. Cohen modified Gödel's function to create structured, but undefined symbols that would ultimately be the sets needed to prove an independence theorem. For the independence of the continuum hypothesis, he introduced the symbols $a_{\delta }$ where $\delta <\aleph _{\tau }$ and $\tau \geq 2.$ Later, each $a_{\delta }$ would become a different subset of $\omega$ so the continuum hypothesis fails in the model. Cohen started by defining the values $F_{\alpha }=\alpha$ for $\alpha \leq \omega .$ Next, $F_{\alpha }$ successively enumerates $a_{\delta }$ for $\delta <\aleph _{\tau },\{a_{\delta },a_{\delta '}\}$ for $\delta <\delta ',$ and $(a_{\delta },a_{\delta '})$ for $\delta <\delta '.$ Let $F_{3\aleph _{\tau }}=\{(a_{\delta },a_{\delta '}):\delta <\delta '\}.$ Finally, $F_{\alpha }$ is defined for $\alpha >3\aleph _{\tau }$ by applying Gödel's set-building operations to $F(\beta )$ where $\beta <\alpha .$ (Cohen 1963, pp. 1144–1145.)
^ Kanamori 2009, p. 65: "Forcing itself went a considerable distance in downgrading any formal theory of classes because of the added encumbrance of having to specify the classes of generic extensions."
^ In the 1960s, this conservative extension theorem was proved independently by Paul Cohen, Saul Kripke, and Robert Solovay. In his 1966 book, Cohen mentioned this theorem and stated that its proof requires forcing. It was also proved independently by Ronald Jensen and Ulrich Felgner, who published his proof in 1971. (Ferreirós 2007, pp. 381–382; Cohen 1966, p. 77; Felgner 1971.)

^ Dauben 1979, p. 66–70. Gray 1994 (p. 828) also gives an example: "… [Cantor's] theorems only deal with sequences that are ordered by a 'law.'" Gray states that "Cantor may have inserted this restriction to avoid problems with Kronecker." However, Ferreirós 2007 (p. 263) corrects Gray: "Like all other authors in the early period of the history of sets, he [Cantor] tended to think of them [sets] as given by a concept or a law, indeed he emphasized this aspect more than other contemporaries." Ferreirós provides examples from Cantor's articles dating from 1872 to 1883.
^ There are $2^{\aleph _{0}}$ points on a line segment, $(2^{\aleph _{0}})^{2}$ points on a plane, and $(2^{\aleph _{0}})^{n}$ points in $n$ -dimensional space. Since $(2^{\aleph _{0}})^{n}=2^{\aleph _{0}\times n}=2^{\aleph _{0}},$ there are the same number of points on a line segment, on a plane, and in $n$ -dimensional space.
^ Quantities and Units - Part 2: Mathematical signs and symbols to be used in the natural sciences and technology, page 15. ISO 80000-2 (ISO/IEC 2009-12-01)
^ This definition of image is used in the proof that NBG's axiom of replacement implies NBG's axiom of separation. See NBG's axiom of separation.
^ This definition of image is used in the proof that NBG's axiom of replacement implies NBG's axiom of separation. See NBG's axiom of separation.
^ Gödel 1940, p. 16; Jech 2003, p. 11; Cunningham 2016, p. 57. Gödel gives definitions 1 and 2 (he uses "over X " instead of "on X " in definition 2). Jech gives all three definitions; Cunningham gives definitions 1 and 3.
^ Cantor, Georg (1874), "Ueber eine Eigenschaft des Inbegriffes aller reellen algebraischen Zahlen", J. Reine Angew. Math., 77: 258–262, doi:10.1515/crll.1874.77.258
^ Johnson, Philip (1972), A History of Set Theory, Prindle, Weber & Schmidt, ISBN 0-87150-154-6
^ Bolzano, Bernard (1975), Berg, Jan (ed.), Einleitung zur Größenlehre und erste Begriffe der allgemeinen Größenlehre, Bernard-Bolzano-Gesamtausgabe, edited by Eduard Winter et al., vol. Vol. II, A, 7, Stuttgart, Bad Cannstatt: Friedrich Frommann Verlag, p. 152, ISBN 3-7728-0466-7 {{citation}}: |volume= has extra text (help)
^ Dauben, Joseph (1979), Georg Cantor: His Mathematics and Philosophy of the Infinite, Harvard University Press, pp. 30–54, ISBN 0-674-34871-0.
^ BIO of Zermelo
^ In Russell's set theory, every Cantorian multiplicity is a set so Russell obtains what he calls Cantor's paradox. Purkert, Walter (1989), "Cantor's Views on the Foundations of Mathematics", in Rowe, David E.; McCleary, John (eds.) (eds.), The History of Modern Mathematics, Volume 1, Academic Press, p. 56, ISBN 0-12-599662-4 {{citation}}: |editor-first2= has generic name (help).
^ Zermelo, Ernst (1930), "Über Grenzzahlen und Mengenbereiche: neue Untersuchungen über die Grundlagen der Mengenlehre" (PDF), Fundamenta Mathematicae, 16: 29–47. English translation: Ewald, William B. (1996), "On boundary numbers and domains of sets: new investigations in the foundations of set theory", From Immanuel Kant to David Hilbert: A Source Book in the Foundations of Mathematics, Oxford University Press, pp. 1208–1233, ISBN 978-0-19-853271-2.
^ Von_Neumann–Bernays–Gödel_set_theory#History.
^ Ferreirós, José (2007), Labyrinth of Thought: A History of Set Theory and Its Role in Mathematical Thought (2nd revised ed.), Birkhäuser, pp. 381–382, ISBN 978-3-7643-8349-7.
^ Blair, Charles E. The Baire category theorem implies the principle of dependent choices. Bull. Acad. Polon. Sci. Sér. Sci. Math. Astronom. Phys. 25 (1977), no. 10, 933–934.
^ Moore 1982, p. 325. Moore's table states that "Principle of Dependent Choices" $\rightarrow$ "Löwenheim–Skolem theorem"—that is, ${\mathsf {DC}}$ implies the Löwenheim–Skolem theorem. The converse is proved in Boolos and Jeffrey 1989, pp. 155–156.
^ Blair, Charles E. The Baire category theorem implies the principle of dependent choices. Bull. Acad. Polon. Sci. Sér. Sci. Math. Astronom. Phys. 25 (1977), no. 10, 933–934.
^ Kanamori 2009, p. 57.
^ Cohen 1966, pp. 107–147. Cohen also gave a detailed proof of Gödel's relative consistency theorems using ZF (Cohen 1966, pp. 85–99).

Gödel's achievement together with the details of his presentation led to the prominence that NBG would enjoy for the next two decades.^[1] Even Paul Cohen's 1963 independence proofs for ZF used tools that Gödel developed for his relative consistency theorems for NBG.^[2] Later, ZFC became more popular than NBG. This was caused by several factors, including the extra work required to handle forcing in NBG,^[a] Cohen's 1966 presentation of forcing, which used ZF,^[3] and the proof that NBG is a conservative extension of ZFC.^[b]

^ Kanamori 2009, p. 65: "Forcing itself went a considerable distance in downgrading any formal theory of classes because of the added encumbrance of having to specify the classes of generic extensions."
^ In the 1960s, this conservative extension theorem was proved independently by Paul Cohen, Saul Kripke, and Robert Solovay. In his 1966 book, Cohen mentioned this theorem and stated that its proof requires forcing. It was also proved independently by Ronald Jensen and Ulrich Felgner, who published his proof in 1971. (Ferreirós 2007, pp. 381–382; Cohen 1966, p. 77; Felgner 1971.)

^ Kanamori 2009, p. 57.
^ Cohen 1963.
^ Cohen 1966, pp. 107–147. Cohen also gave a detailed proof of Gödel's relative consistency theorems using ZF (Cohen 1966, pp. 85–99).

Add to ZFC

ZFC

May not be an error! See long debate about axiom of infinity in Talk!!

2nd justification of omitting an existence axiom by the axiom of infinity asserting the existence of an infinite set. However, the statement of the axiom of infinity assumes the existence of the empty set! Can only do this with vN's axiom of infinity.

Error in Zermelo set theory: The aim of Zermelo's paper

Zermelo set theory

The introduction states that the very existence of the discipline of set theory "seems to be threatened by certain contradictions or "antinomies", that can be derived from its principles – principles necessarily governing our thinking, it seems – and to which no entirely satisfactory solution has yet been found". Zermelo is of course referring to the "Russell antinomy".

In 1908, Zermelo published his axiom system for set theory. He had two motivations for developing the axiom system: eliminating the paradoxes and securing his proof of the well-ordering theorem.^[1] Zermelo had proved this theorem in 1904 using the axiom of choice, but his proof was criticized for a variety of reasons.^[2] His response to the criticism included his axiom system and a new proof of the well-ordering theorem. His axioms support this new proof, and they eliminate the paradoxes by restricting the formation of sets.^[3]

Locke, Descartes, Spinoza, Leibnitz quotes

God is the only thing I positively conceive as infinite. As to other things like the extension of the world and the number of parts into which matter is divisible, I confess I do not know whether they are absolutely infinite; I merely know that I can see no end to them. . . .5 [Descartes, letter to More, 5 February 1649, in Philosophical Letters, trans., ed. Anthony Kenny (Minneapolis: University of Minnesota Press, 1981), p. 242.]

René Descartes, Principles of Philosophy, p. 7: Baruch Spinoza, Ethics Demonstrated in Geometrical Order, p. 1:

Gutenberg's Principles OF THE PRINCIPLES OF HUMAN KNOWLEDGE. XXIV. That in passing from the knowledge of God to the knowledge of the creatures, it is necessary to remember that our understanding is finite, and the power of God infinite.

Descartes, René (2012) [1644], Jonathan Bennett (ed.), Principles of Philosophy (PDF).
Leibniz, Gottfried Wilhelm (1896), New essays Concerning Human Understanding (PDF), translated by Langley, Alfred Gideon, London: The Macmillan Company.
Locke, John (1824) [1690], "An Essay concerning Human Understanding", The Works of John Locke, London: C. Baldwin, printer, pp. 104–418.

Spinoza, Baruch (2004) [completed 1674, published 1677], Jonathan Bennett (ed.), Ethics Demonstrated in Geometrical Order (PDF).

Baruch Spinoza, Ethics Demonstrated in Geometrical Order, p. 1:

Gottfried Wilhelm Leibniz, C New essays Concerning Human Understanding] p. 162: "The true infinite exists, strictly speaking, only in the absolute, which is anterior to all composition, and is not formed by the additions of parts."

163 Th. [I do not find that it has been established that the con sideration of the finite and the infinite takes place wherever there is bulk and magnitude. And the true infinite is not a modification, it is the absolute ; on the contrary, when it is modified, it is limited and forms a finite.]

The idea of the absolute is in us internally, like that of being ; these absolutes are nothing else than the attributes of God,

Leibniz stated: "The true infinite exists, strictly speaking, only in the absolute, which is anterior to all composition, and is not formed by the additions of parts."

Spinoza used absolute infinity in his definition of God: "By ‘God’ I understand: a thing that is absolutely infinite, i.e. a substance consisting of an infinity of attributes, each of which expresses an eternal and infinite essence."

Baruch Spinoza, Ethics Demonstrated in Geometrical Order, p. 1:

By ‘God’ I understand: a thing that is absolutely infinite, i.e. a substance consisting of an infinity of attributes, each of which expresses an eternal and infinite essence.

René Descartes, Principles of Philosophy, p. 7: 24. In passing from knowledge of God to knowledge of his creation, we should bear in mind that he is infinite and we are finite.

Locke, An Essay concerning Human Understanding

CHAP. XV, §12, p. 188. What I say of man I say of all finite beings; who, though they may far exceed man in knowledge and power, yet are no more than the meanest creature, in comparison with God himself. Finite of any magnitude holds not any proportion to infinite. God’s infinite duration being accompanied with infinite knowledge and infinite power, he sees all things past and to come; and they are no more distant from his knowledge, no farther removed from his sight, than the present: they all lie under the same view; and there is nothing which he cannot make exist each moment he pleases.

CHAP. XVII.: Of Infinity, § 1, p. 194. It is true, that we cannot but be assured, that the great God, of whom and from whom are all things, is incomprehensibly infinite: but yet when we apply to that first and supreme being our idea of infinite, in our weak and narrow thoughts, we do it primarily in respect to his duration and ubiquity; and, I think, more figuratively to his power, wisdom, and goodness, and other attributes, which are properly inexhaustible and incomprehensible, &c. For, when we call them infinite, we have no other idea of this infinity, but what carries with it some reflection on, and imitation of, that number or extent of the acts or objects of God’s power, wisdom, and goodness, which can never be supposed so great or so many, which these attributes will not always surmount and exceed, let us multiply them in our thoughts as far as we can, with [195] all the infinity of endless number.

CHAP. XVII, § 13, p. 202. Though it be hard, I think, to find any one so absurd as to say, he has the positive idea of an actual infinite number; the infinity whereof lies only in a power still of adding any combination of units to any former number, and that as long and as much as one will; the like also being in the infinity of space and duration, which power leaves always to the mind room for endless additions; yet there be those who imagine they have positive ideas of infinite duration and space. It would, I think, be enough to destroy any such positive idea of infinite, to ask him that has it, whether he could add to it or no; which would easily show the mistake of such a positive idea. We can, I think, have no positive idea of any space or duration which is not made up, and commensurate to repeated numbers of feet or yards, or days and years, which are the common measures, whereof we have the ideas in our minds, and whereby we judge of the greatness of this sort of quantities. And therefore, since an infinite idea of space or duration must needs be made up of infinite parts, it can have no other infinity than that of number, capable still of farther addition: but not an actual positive idea of a number infinite. For, I think, it is evident that the addition of finite things together (as are all lengths, whereof we have the positive ideas) can never otherwise produce the idea of infinite, than as number does; which consisting of additions of finite units one to another, suggests the idea of infinite, only by a power we find we have of still increasing the sum, and adding more of the same kind, without coming one jot nearer the end of such progression.

Locke It is true, that we cannot but be assured, that the great God, of whom and from whom are all things, is incomprehensibly infinite: but yet when we apply to that first and supreme being our idea of infinite, in our weak and narrow thoughts, we do it primarily in respect to his duration and ubiquity;

CHAP. XVII, § 16, p. 205. whether any one has or can have a positive idea of an actual infinite number, I leave him to consider, till his infinite number be so great that he himself can add no more to it; and as long as he can increase it, I doubt he himself will think the idea he hath of it a little too scanty for positive infinity.

René Descartes, The Principles of Philosophy

p. : XXIV. That in passing from the knowledge of God to the knowledge of the creatures, it is necessary to remember that our understanding is finite, and the power of God infinite.

But as we know that God alone is the true cause of all that is or can be, we will doubtless follow the best way of philosophizing, if, from the knowledge we have of God himself, we pass to the explication of the things which he has created, and essay to deduce it from the notions that are naturally in our minds, for we will thus obtain the most perfect science, that is, the knowledge of effects through their causes. But that we may be able to make this attempt with sufficient security from error, we must use the precaution to bear in mind as much as possible that God, who is the author of things, is infinite, while we are wholly finite.

p. : XXVI. That it is not needful to enter into disputes [Footnote: "to essay to comprehend the infinite."—FRENCH.] regarding the infinite, but merely to hold all that in which we can find no limits as indefinite, such as the extension of the world, the divisibility of the parts of matter, the number of the stars, etc.

We will thus never embarrass ourselves by disputes about the infinite, seeing it would be absurd for us who are finite to undertake to determine anything regarding it, and thus as it were to limit it by endeavouring to comprehend it. We will accordingly give ourselves no concern to reply to those who demand whether the half of an infinite line is also infinite, and whether an infinite number is even or odd, and the like, because it is only such as imagine their minds to be infinite who seem bound to entertain questions of this sort. And, for our part, looking to all those things in which in certain senses, we discover no limits, we will not, therefore, affirm that they are infinite, but will regard them simply as indefinite. Thus, because we cannot imagine extension so great that we cannot still conceive greater, we will say that the magnitude of possible things is indefinite, and because a body cannot be divided into parts so small that each of these may not be conceived as again divided into others still smaller, let us regard quantity as divisible into parts whose number is indefinite; and as we cannot imagine so many stars that it would seem impossible for God to create more, let us suppose that their number is indefinite, and so in other instances.

René Descartes, Principles of Philosophy

p. 7: 24. In passing from knowledge of God to knowledge of his creation, we should bear in mind that he is infinite and we are finite. Since God alone is the true cause of everything that does or could exist, it’s clear that the best way to go about philosophizing [here = ‘doing philosophy or natural science’] is to •start from what we know of God himself and •try to derive from that knowledge an explanation of the things created by him. That’s the way to acquire the most perfect scientific knowledge, i.e. knowledge of effects through their causes. To minimize our chances of going wrong in this process, we must carefully bear in mind •that God, the creator of all things, is infinite, and •that we are altogether finite.

26. We should steer clear of arguments about the infinite. When we see something as unlimited—e.g. the extension of the world, the division of the parts of matter, the number of the stars, and so on—we should regard it ·not as infinite but· as indefinite. That will spare us tiresome arguments about the infinite. Given that we are finite, it would be absurd for us to ·try to· establish any definite results concerning the infinite, because that would be trying to limit it and get our minds around it. When questions such as these are asked: Would half an infinite line also be infinite? Is an infinite number odd or even? we shan’t bother to answer. No-one has any business thinking about such matters, it seems to me, unless he thinks his own mind is infinite! What we’ll do is this: faced with something that so far as we can see is unlimited in some respect, we’ll describe it not as ‘infinite’ but as ‘indefinite’. •An example: we can’t imagine a size so big that we can’t conceive of the possibility of a bigger; so our answer to the question ‘How big could a thing be?’ should be ‘Indefinitely big’. •Another: however many parts a given body is divided into, we can still conceive of each of those parts as being further divisible; so our answer to the question ‘How many parts can a body be divided into?’ is ‘Indefinitely many’. •A third: no matter how numerous we imagine the stars to be, we think that God could have created even more; so we’ll suppose that there’s an indefinite number of stars. And the same will apply in other cases. 27. The difference between the indefinite and the infinite. The point of using ‘indefinite’ rather than ‘infinite’ is to reserve ‘infinite’ for God, because he’s the only thing that our understanding •positively tells us doesn’t have any limits. The most we know about anything else is the •negative information that we can’t find any limits in it. 28. It’s not the •final but the •efficient causes of created things that we must investigate. [In contemporary terms, that is equivalent to saying ‘What we must investigate are not created things’ •purposes but their •causes’.] We’ll never explain natural things in terms of the purposes that God or nature may have had when creating them, [added in the French] and we shall entirely banish them from our natural science. Why? Because we shouldn’t be so arrogant as to think that we can share in God’s plans. We should bring

Baruch Spinoza, Ethics Demonstrated in Geometrical Order

p. 1: By ‘God’ I understand: a thing that is absolutely infinite, i.e. a substance consisting of an infinity of attributes, each of which expresses an eternal and infinite essence. I say ‘absolutely infinite’ in contrast to ‘infinite in its own kind’. If something is infinite only in its own kind, there can be attributes that it doesn’t have; but if something is absolutely infinite its essence ·or nature· contains every positive way in which a thing can exist—·which means that it has all possible attributes.

p. 6: A substance that is absolutely infinite is indivisible.

Also, $\omega$ is neither even nor odd, which shows that Descartes' question of "whether an infinite number is even or odd" is irrelevant to Cantor's work.

By "determination", Cantor meant can be determined or understood by humans (for example, through mathematics). An argument relevant to Cantor's work appears in Locke: "It would, I think, be enough to destroy any such positive idea of infinite, to ask him that has it, whether he could add to it or no; which would easily show the mistake of such a positive idea." If there were an infinite number greater than all the numbers and if adding to it creates a larger number, then the original infinite number is not the greatest number. On the other hand, if you cannot add to the infinite number, then it lacks a fundamental property of numbers so it cannot be a number.

Thus, some mathematicians and philosophers before Cantor distinguished between the finite and the absolute, which was not reachable from below and nothing could be added to it. Also, they associated the absolute with God. Cantor introduced the transfinite so his division consisted of the finite, the transfinite, and absolute infinity. He also associated the absolute with God. In his Grundlagen, Cantor stated that the sequence of all finite and transfinite numbers form an absolutely infinite sequence. So he accepted the need for an absolute infinity. However, he did not state that assigning this absolutely infinite sequence an order type (which is equivalent in modern set theory to assigning it an ordinal) leads to a contradiction. However, in a 1897 letter, he stated that he knew this when he was writing his Grundlagen.

Puckert argues that Cantor did know that it is inconsistent to view the totality of ordinals as a set (p. 57)

He found some of the arguments against infinite numbers to be flawed Cantor then introduced the transfinite "which, just like the finite, can be determined by well-defined and distinguishable numbers."^[4] The transfinite follows the finite and comes before the Absolute. Philosophers had introduced an Absolute that was different from the finite because they had developed arguments demonstrating that infinite numbers led to contradictions and because the infinite was traditionally associated with God. Cantor needed an Absolute for nearly the same reasons.

Locke, An Essay concerning Human Understanding

CHAP. XV, §12, p. 188. What I say of man I say of all finite beings; who, though they may far exceed man in knowledge and power, yet are no more than the meanest creature, in comparison with God himself. Finite of any magnitude holds not any proportion to infinite. God’s infinite duration being accompanied with infinite knowledge and infinite power, he sees all things past and to come; and they are no more distant from his knowledge, no farther removed from his sight, than the present: they all lie under the same view; and there is nothing which he cannot make exist each moment he pleases.

CHAP. XVII.: Of Infinity, § 1, p. 194. It is true, that we cannot but be assured, that the great God, of whom and from whom are all things, is incomprehensibly infinite: but yet when we apply to that first and supreme being our idea of infinite, in our weak and narrow thoughts, we do it primarily in respect to his duration and ubiquity; and, I think, more figuratively to his power, wisdom, and goodness, and other attributes, which are properly inexhaustible and incomprehensible, &c. For, when we call them infinite, we have no other idea of this infinity, but what carries with it some reflection on, and imitation of, that number or extent of the acts or objects of God’s power, wisdom, and goodness, which can never be supposed so great or so many, which these attributes will not always surmount and exceed, let us multiply them in our thoughts as far as we can, with [195] all the infinity of endless number.

CHAP. XVII, § 13, p. 202. Though it be hard, I think, to find any one so absurd as to say, he has the positive idea of an actual infinite number; the infinity whereof lies only in a power still of adding any combination of units to any former number, and that as long and as much as one will; the like also being in the infinity of space and duration, which power leaves always to the mind room for endless additions; yet there be those who imagine they have positive ideas of infinite duration and space. It would, I think, be enough to destroy any such positive idea of infinite, to ask him that has it, whether he could add to it or no; which would easily show the mistake of such a positive idea. We can, I think, have no positive idea of any space or duration which is not made up, and commensurate to repeated numbers of feet or yards, or days and years, which are the common measures, whereof we have the ideas in our minds, and whereby we judge of the greatness of this sort of quantities. And therefore, since an infinite idea of space or duration must needs be made up of infinite parts, it can have no other infinity than that of number, capable still of farther addition: but not an actual positive idea of a number infinite. For, I think, it is evident that the addition of finite things together (as are all lengths, whereof we have the positive ideas) can never otherwise produce the idea of infinite, than as number does; which consisting of additions of finite units one to another, suggests the idea of infinite, only by a power we find we have of still increasing the sum, and adding more of the same kind, without coming one jot nearer the end of such progression.

René Descartes, The Principles of Philosophy

p. : XXIV. That in passing from the knowledge of God to the knowledge of the creatures, it is necessary to remember that our understanding is finite, and the power of God infinite.

But as we know that God alone is the true cause of all that is or can be, we will doubtless follow the best way of philosophizing, if, from the knowledge we have of God himself, we pass to the explication of the things which he has created, and essay to deduce it from the notions that are naturally in our minds, for we will thus obtain the most perfect science, that is, the knowledge of effects through their causes. But that we may be able to make this attempt with sufficient security from error, we must use the precaution to bear in mind as much as possible that God, who is the author of things, is infinite, while we are wholly finite.

p. : XXVI. That it is not needful to enter into disputes [Footnote: "to essay to comprehend the infinite."—FRENCH.] regarding the infinite, but merely to hold all that in which we can find no limits as indefinite, such as the extension of the world, the divisibility of the parts of matter, the number of the stars, etc.

We will thus never embarrass ourselves by disputes about the infinite, seeing it would be absurd for us who are finite to undertake to determine anything regarding it, and thus as it were to limit it by endeavouring to comprehend it. We will accordingly give ourselves no concern to reply to those who demand whether the half of an infinite line is also infinite, and whether an infinite number is even or odd, and the like, because it is only such as imagine their minds to be infinite who seem bound to entertain questions of this sort. And, for our part, looking to all those things in which in certain senses, we discover no limits, we will not, therefore, affirm that they are infinite, but will regard them simply as indefinite. Thus, because we cannot imagine extension so great that we cannot still conceive greater, we will say that the magnitude of possible things is indefinite, and because a body cannot be divided into parts so small that each of these may not be conceived as again divided into others still smaller, let us regard quantity as divisible into parts whose number is indefinite; and as we cannot imagine so many stars that it would seem impossible for God to create more, let us suppose that their number is indefinite, and so in other instances.

René Descartes, Principles of Philosophy

p. 7: 24. In passing from knowledge of God to knowledge of his creation, we should bear in mind that he is infinite and we are finite. Since God alone is the true cause of everything that does or could exist, it’s clear that the best way to go about philosophizing [here = ‘doing philosophy or natural science’] is to •start from what we know of God himself and •try to derive from that knowledge an explanation of the things created by him. That’s the way to acquire the most perfect scientific knowledge, i.e. knowledge of effects through their causes. To minimize our chances of going wrong in this process, we must carefully bear in mind •that God, the creator of all things, is infinite, and •that we are altogether finite.

26. We should steer clear of arguments about the infinite. When we see something as unlimited—e.g. the extension of the world, the division of the parts of matter, the number of the stars, and so on—we should regard it ·not as infinite but· as indefinite. That will spare us tiresome arguments about the infinite. Given that we are finite, it would be absurd for us to ·try to· establish any definite results concerning the infinite, because that would be trying to limit it and get our minds around it. When questions such as these are asked: Would half an infinite line also be infinite? Is an infinite number odd or even? we shan’t bother to answer. No-one has any business thinking about such matters, it seems to me, unless he thinks his own mind is infinite! What we’ll do is this: faced with something that so far as we can see is unlimited in some respect, we’ll describe it not as ‘infinite’ but as ‘indefinite’. •An example: we can’t imagine a size so big that we can’t conceive of the possibility of a bigger; so our answer to the question ‘How big could a thing be?’ should be ‘Indefinitely big’. •Another: however many parts a given body is divided into, we can still conceive of each of those parts as being further divisible; so our answer to the question ‘How many parts can a body be divided into?’ is ‘Indefinitely many’. •A third: no matter how numerous we imagine the stars to be, we think that God could have created even more; so we’ll suppose that there’s an indefinite number of stars. And the same will apply in other cases. 27. The difference between the indefinite and the infinite. The point of using ‘indefinite’ rather than ‘infinite’ is to reserve ‘infinite’ for God, because he’s the only thing that our understanding •positively tells us doesn’t have any limits. The most we know about anything else is the •negative information that we can’t find any limits in it. 28. It’s not the •final but the •efficient causes of created things that we must investigate. [In contemporary terms, that is equivalent to saying ‘What we must investigate are not created things’ •purposes but their •causes’.] We’ll never explain natural things in terms of the purposes that God or nature may have had when creating them, [added in the French] and we shall entirely banish them from our natural science. Why? Because we shouldn’t be so arrogant as to think that we can share in God’s plans. We should bring

Baruch Spinoza, Ethics Demonstrated in Geometrical Order

p. 1: By ‘God’ I understand: a thing that is absolutely infinite, i.e. a substance consisting of an infinity of attributes, each of which expresses an eternal and infinite essence. I say ‘absolutely infinite’ in contrast to ‘infinite in its own kind’. If something is infinite only in its own kind, there can be attributes that it doesn’t have; but if something is absolutely infinite its essence ·or nature· contains every positive way in which a thing can exist—·which means that it has all possible attributes.

p. 6: A substance that is absolutely infinite is indivisible.

p. 8: If corporeal substance is infinite, they say, let us conceive it to be divided into two parts. If each part is finite, then an infinite is composed of two finite parts, which is absurd. If each part is infinite, then there is one infinite twice as large as another, which is also absurd. •Again, if an infinite quantity is measured by parts each equal to a foot, it will consist of infinitely many of them, as it will also if it is measured by parts each equal to an inch. So one infinite number will be twelve times as great as another, which is no less absurd.

Gottfried Wilhelm Leibniz, New essays Concerning Human Understanding p. 162: "The true infinite exists, strictly speaking, only in the absolute, which is anterior to all composition, and is not formed by the additions of parts."

163 Th. [I do not find that it has been established that the con sideration of the finite and the infinite takes place wherever there is bulk and magnitude. And the true infinite is not a modification, it is the absolute ; on the contrary, when it is modified, it is limited and forms a finite.]

The idea of the absolute is in us internally, like that of being ; these absolutes are nothing else than the attributes of God,

p. 155 Cf. ante, pp. 16, 17; also New Essays, Bk. II., chap. 17, 1. The proof that the universe is not, strictly speaking, a whole, is given in the letter to Des Bosses, March 11, 1706, Gerliardt, Vol. 2, p.304sg.,Erdmann, pp. 435-436. TR.

ADD MORE LEIBNIZ!!!

i] ON HUMAN UNDERSTANDING 17

a space, a time, or a number, there is always another greater than it without end; and that thus the true infinite is not found in a whole composed of parts. It is none the less, how ever, found elsewhere ; namely, in the absolute, which is with out parts, and which has influence over compound things, because they result from the limitation of the absolute. The positive infinite, then, being nothing else than the absolute, it may be said that there is in this sense a positive idea of the infinite, and that it is anterior to that of the finite. For the rest, in rejecting a composite infinite, we do not deny the demonstrations of the geometers de Seriebus infinitis, and par ticularly what the excellent Mr. Newton has given us, not to mention my own contributions to the subject.

1. Ph. One of the most important notions is that of the finite and the infinite, which are regarded as modes of quantity.

Th. [Properly speaking, it is true that there is an infinite number of things, i.e. that there are always more of them than can be assigned. But there is no infinite number, neither line nor other infinite quantity, if these are understood as veritable wholes, as it is easy to prove. The schools have meant or have been obliged to say that, in admitting a syncategorematic in- M

162 LEIBNITZ S CRITIQUE OF LOCKE [BK. n

finite, 1 as they call it, and not a categorematic infinite. The true infinite exists, strictly speaking, only in the absolute, which is anterior to all composition, and is not formed by the additions of parts. 2 ]

163

The idea of the absolute is in us internally, like that of being ; these absolutes are nothing else than the attributes of God,

Ph. When we apply our idea of the infinite to the first Being, we do it primarily in respect to his duration and ubi quity, and, more figuratively, to his power, his wisdom, his goodness, and his other -attributes.

Th. [Not more figuratively, but less immediately, because the other attributes make their importance known through relation to those into which enters the consideration of parts.]

2. Ph. I thought it was established that the mind regards the finite and the infinite as modifications of extension 3 and duration.

Th. [I do not find that it has been established that the con sideration of the finite and the infinite takes place wherever there is bulk and magnitude. And the true infinite is not a modification, it is the absolute ; on the contrary, when it is modified, it is limited and forms a finite.]

__ mentions Cantor's knowledge of philosophy, especially relating to the question of infinite number.

I believe, indeed, with Mr. Locke that, properly speaking, we may say that there is no space, time, nor number which is infinite, but that it is only true that however great ^ may be 1 Gerhardt's test seems here, for some reason, to be defective. It reads thus : "Mais qu'il est seuloment vray que pour t;rand que Uiy sans fin," etc. Erdmann's seems the more correct, and is therefore followed in the translation! It reads thus: "Mais qu'il est seulement vrai que pour grand que soit uii espace, un terns, ou un nomhre, il y en a toujours un autre plus grand que lui

I] ON HUMAN UNDERSTANDING 17 a space, a time, or a number, there is always another greater than it without end ; and that thus the true infinite is not found in a whole composed of parts. It is none the less, however, found elsewhere; namely, in the absolute, which is without parts, and which has influence over compound things, because they result from the limitation of the absolute. The positive infinite, then, being nothing else than the absolute, it may be said that there is in this sense a positive idea of the infinite, and that it is anterior to that of the finite.

p. 158: Th. [But in order to draw from them the notion of eternity, it is necessary to think besides that the same reason always exists for going farther. It is this rational consideration which achieves the notion of the infinite or the indefinite in possible progress. Thus the senses alone cannot suffice to cause the formation of these notions. And ultimately it may be said that the idea of the absolute 1' is anterior in the nature of things to that of the limits which are added, but we notice the former only as we commence with what is limited and strikes our senses.]

1 The idea of the absolute belongs to our reason as such, cf. New Essays, Bk. II., chap. 17, § 3, Th., § 16, Th., though we first hecorae aware of it througli our consciousness o£ the particular ideas o£ the reason as limitations of the idea of the absolute. —Tr.

p. 161-2: § 1. Ph. One of the most important notions is that of the finite and the infinite, which are regarded as modes of quantity. Th. [Properly speaking, it is true that there is an infinite number of things, i.e. that there are always more of them than can be assigned. But there is no infinite number, neither line nor other infinite quantity, if these are understood as veritable wholes, as it is easy to prove. The schools have meant or have been obliged to say that, in admitting a syncategorematic in-finite/ as they call it, and not a categorematic infinite. The true infinite exists, strictly speaking, only in the absolute, which is anterior to all composition, and is not formed by the additions of parts. ^] Cf. ante, pp. 16, 17 ; also New Essays, Bk. II., chap. 13, § 21.—Tr.

INDEX: Absolute, idea of, anterior to that of limits, 158; anterior to all composition, 102 ; is that of the infinite, 12, 17, 162; "in us internally," 163; opposed to relative, 2.36.

These mathematicians and philosophers distinguished between the finite and the absolute. They also stated that the absolute was not reachable from below and nothing could be added to it. Also, they associated the absolute with God. Cantor introduced the transfinite so his division consisted of the finite, the transfinite, and absolute infinity. He also associated the absolute with God. In his Grundlagen, Cantor stated that the sequence of all finite and transfinite numbers form an absolutely infinite sequence. So he accepted the need for an absolute infinity. However, he did not state that assigning this absolutely infinite sequence an order type (which is equivalent in modern set theory to assigning it an ordinal) leads to a contradiction. However, in a 1897 letter, he stated that he knew this when he was writing his Grundlagen.

"He [Cantor] regarded ... the relation between transfinite and Absolute as analogous to that between the finite and infinity." p. 65, Ferreiros 2004 "Motives"

experiment

$(x,y)\,{\xrightarrow[{ProdV}]{}}\,((x,y),z)=(x,y,z)$

$(x,y)\,{\xrightarrow[{ProdV}]{}}\,(x,y,z)\,{\xrightarrow[{CPerm}]{}}\,(z,x,y)$

$(x,y)\,{\xrightarrow[{ProdV}]{}}\,(x,y,z)\,{\xrightarrow[{Trans}]{}}\,(x,z,y)$

$(x,y)\,{\xrightarrow[{ProdV}]{}}\,(x,y,z)\,{\xrightarrow[{Trans}]{}}\,(x,z,y)\,{\xrightarrow[{CPerm}]{}}\,(y,x,z)\,{\xrightarrow[{Dom}]{}}\,(y,x)$ SHORTER???? }}

Possible note for Cantor's first set theory article?

See: Bell 1937, pp. 568–569; Hardy and Wright 1938, p. 159 (6th ed., pp. 205–206); Birkhoff and Mac Lane 1941, p. 392, (5th ed., pp. 436–437); Spivak 1967, pp. 368–370, 528 (4th ed., pp. 448 - 1?–449, ??).

PROBLEM: Hardy and Wright give 3 examples: non-constructive, Liouville, particular trans # (e and pi). Note: The three textbooks give a proof presentation similar to Perron's: they give the non-constructive proof, point out that it is non-constructive, and compare it to Liouville's constructive existence proof. This proof presentation appeared in the 192_ second edition of Hardy's ____________ together with a reference to Perron's 1921 book. Also, Hardy and Wright reference Perron's 1921 book, while Birkhoff and Mac Lane, and Spivak reference Hardy and Wright's book. See: Bell 1937, pp. 568–569; Hardy and Wright 1938, p. 159 (6th ed., pp. 205–206); Birkhoff and Mac Lane 1941, p. 392, (5th ed., pp. 436–437); Spivak 1967, pp. 368–370, 528 (4th ed., pp. 447??–449, ??).

Gödel's work with proper classes

Gödel uses proper classes throughout his 1940 monograph.^[5] Below are two examples of his work. First, some definitions: If $(A,W)$ is a class well-ordered by $W,$ its proper initial segments are the classes $\{x\in A:xWy\}$ where $y\in A.$ If $B\subseteq A,$ then $Inf(B)$ is the least element of $B.\;Ord$ is the class of all ordinals.

Theorem. Every well-ordered proper class whose proper initial segments are sets is order isomorphic to $Ord$ .

Outline of proof. Let $A$ be a proper class that is well-ordered by $W.$ Define the order isomorphism $F:(Ord,<)\rightarrow (A,W)$ by transfinite recursion: $F(\alpha )=Inf(A\setminus \{F(\beta ):\beta <\alpha \}).$ Note that $\{F(\beta ):\beta <\alpha \}$ is a proper initial segment of $A.$

^[6]

Constructing the constructible sets. Gödel's strategy is to modify the construction used in the proof of the class existence theorem so that it only builds sets from previously constructed sets. To do this, he takes the pairing axiom and the class axioms (see Class axioms used in this construction and relativizes them to sets that were previously constructed. For example, the domain axiom, which given a class Y of n-tuples produces Dom(Y), the domain of (n-1)-tuples of class Y, becomes the operation F_3(X, Y) = X \cap Dom(Y) where the sets X and Y were previously constructed.; he also adds one operations that creates the set consisting of all previously constructed sets. To define "previously constructed," he uses ordinals to order the stages of the construction.

Since each fundamental operation has two set arguments and there are eight fundamental operations and one special operation, he defines a well-ordering $S$ on $9\times Ord\times Ord$ to order the stages of his construction. His theorem produces an order isomorphism $J:(Ord,<)\rightarrow (9\times Ord\times Ord,S),$ which he uses to define a function $F$ on $Ord$ that builds the constructible sets. For example, if $J(\gamma )=(1,\alpha ,\beta ),$ then $F(\gamma )=\{F(\alpha ),F(\beta )\}$ since the pairing operation is operation 1. Also, $\alpha <\gamma$ and $\beta <\gamma ,$ so $F(\alpha )$ and $F(\beta )$ have been previously constructed. Gödel defines the constructible universe $L$ as the range of $F.$ ^[7] Since $F$ maps $Ord$ onto $L,$ it produces a well-ordering of $L$ , which implies the axiom of global choice (see Implications of the axiom of limitation of size). Since $L$ is a model of NBG that was built using NBG – {axiom of global choice}, the axiom of global choice is relatively consistent with NBG – {axiom of global choice}.

Talk for Cantor's first set theory article: Connections between books that claim Cantor's proof is non-constructive

The three textbooks follow Perron's format: they give the non-constructive existence proof and compare it to (??) Liouville's constructive existence proof. Hardy and Wright reference Perron's book, while Birkhoff and Mac Lane reference Hardy and Wright's book. Spivak includes Hardy and Wright's book in his suggested reading and states "______________________________". [Hardy and Wright 1938, p. 159 (6th ed., pp. 205–206); Birkhoff and Mac Lane 1941, p. 392, (5th ed., pp. 436–437); Spivak 1967, pp. 369–370 (4th ed., pp. 448–449)].
Bell does not reference any books in his Men of Mathematics. However, Bell admired Hardy's work and Hardy visited the CalTech where Bell taught in 193_. Bell 1937, pp. 568–569;

^[8]

In the section "The disagreement about Cantor's existence proof", several quite successful books are mentioned that claim Cantor's proof of the existence of transcendentals is non-constructive. It turns out that these authors' opinions about Cantor's proof can be traced to Oskar Perron's 1921 book Irrationalzahlen (Irrational Numbers). This book gives the non-constructive proof, points out that it is non-constructive, and compares it to Liouville's constructive existence proof. Below, we say that a book "gives a proof presentation similar to Perron's" if it gives the non-constructive proof, points out that it is non-constructive, and compares it to Liouville's constructive existence proof. MENTION HOW SUCCESSFUL PERRON'S Book was. Here are some books that were directly and indirectly influenced by Perron's book:

In 1924, G. H. Hardy's 2nd edition of Orders of Infinity: The 'Infinitärcalcül' of Paul Du Bois-Reymond appeared. It references Perron's book and gives a proof presentation similar to Perron's. The first edition of Hardy's book appeared in 1910 and does not contain the non-constructive proof.

In 1938, G. H. Hardy and E. M. Wright's An Introduction to the Theory of Numbers appeared. It references Perron's book and gives a proof presentation similar to Perron's.

In 1941, Garrett Birkhoff and Saunders MacLane's A Survey of Modern Algebra appeared. It references Hardy and Wright's book and gives a proof presentation similar to Perron's.

In 1967, Michael Spivak's Calculus appeared. It references Hardy and Wright's book and gives a proof presentation similar to Perron's.

In 1937, E. T. Bell's Men of Mathematics appeared, which states that Cantor's proof is ____________________DIRECT QUOTE??. Unfortunately, Bell's book contains no references. However, reading MENTION CONSTANCE REID'S BOOK!! So it is quite possible that Bell's opinion that Cantor's proof is non-constructive came from Hardy's 1924 edition of Orders of Infinity: The 'Infinitärcalcül' of Paul Du Bois-Reymond or directly from Hardy himself.

In 1923, A. Fraenkel's 2nd edition of Einleitung in die Mengenlehre (Introduction to Set Theory) contains a paragraph that gives Cantor's constructive proof and points out that it is constructive. The first edition of Fraenkel's book appeared in 1919 without this paragraph. Both editions contain an earlier paragraph having the non-constructive proof that is neither pointed out to be non-constructive nor is it ascribed to Cantor (CHECK THIS!!). The appearance of a new paragraph containing Cantor's constructive proof in Fraenkel's 1923 edition seems to suggest that he was either aware of Perron's 1921 book or was reacting to the opinions of other mathematicians who may have been influenced by Perron's book. Both mathematicians were at German universities at this time. By the way, I use Fraenkel's 1930 quote in the Wikipedia article because it goes beyond his 1923 book: his 1930 quote points out there is a "widespread interpretation" that claims that Cantor's proof is non-constructive.

Future

Hi Michael, I just want to thank you for the encouragement and support you have given me since I first started rewriting the "Cantor's first uncountability proof" article years ago.

Mention how GA nomination greatly improved my Wikipedia writing. I now understand how to write leads and have learned where Wikipedia keeps its writing rules.

I know that you like to convince mathematicians to write for Wikipedia. Well, you've succeeded with this amateur mathematician. I now work fairly regularly on Wikipedia.

I just wanted to say how much I enjoy writing for Wikipedia. It has motivated me to look deeply at various subjects in math, history of math, and philosophy of math. I find that I learn a lot more if I take the time to learn a subject well enough to explain it to Wikipedia readers. Usually, I just add or rewrite sections to existing articles (for example, I rewrote the Georg Cantor section titled "Paradoxes of set theory" so it now is Absolute infinite, well-ordering theorem, and paradoxes). Also, you spotted my rewrite of a section in Controversy over Cantor's theory. I appreciated your "Thank you" for my work on that article. I just completed my rewriting of Axiom of limitation of size. I have several new writing projects.

Talk: Is von Neumann's 1923 axiom weaker than or equivalent to the axiom of limitation of size?

I have not found the answer to this question in the literature nor have I been able to answer it myself. Let N⁻ be von Neumann's 1925 axiom system − {axiom of limitation of size} + {his 1923 axiom} + {axiom of regularity}, which is equivalent to NBG − {axioms of global choice and replacement} + {his 1923 axiom}. Proving that his 1923 axiom is equivalent to the axiom of limitation of size can be done by proving the axiom of replacement from N⁻. Then since replacement and his 1923 axiom implies global choice, we can prove the axiom of limitation of size (regularity, which von Neumann stated but did not add to his axiom system, is required here). One difficulty is that N⁻ lacks the axioms of separation and choice, so you don't have many axioms to work with. Proving that his 1923 axiom is weaker than the axiom of limitation of size can be done by exhibiting a model of N⁻ in which either the axiom of separation, replacement, or global choice fails.

Wikipedia can be more accurate than some books

I learned about the lack of accuracy in some books through my work on The disagreement about Cantor's existence proof. Here we have a case of proof by repeated assertion. The notion that Cantor's proof of the existence of transcendentals is non-constructive seems to have first appeared in Perron's book on irrational numbers. Then Hardy and Wright got it from Perron's book (they reference his book), then Birkhoff and MacLane and later Spivak got it from Hardy and Wright's book (they reference Hardy and Wright). From these successful books, it has spread very widely.

Gödel's L and the axiom of limitation of size

The proof of Theorem 2, | V_κ | = κ, for the models V_κ where κ is a strongly inaccessible cardinal proof does not generate an explicit function from κ to V_κ. This is because the α+1 case depends on 2^λ being well-ordered so that 2^λ < κ makes sense. Since the axiom of choice only specifies the existence of a well-ordering, this proof does not generate an explicit function from κ to V_κ.

In his 1940 monograph on the relative consistency of the axiom of choice and the generalized continuum hypothesis, Gödel starts with a model V of NBG without the axiom of global choice. He uses tranfinite recursion to define a function F(α) that builds one set for each ordinal. He then defines the class L of constructible sets as the domain of F(α). He then proves that L is a model of NBG and that F(α) builds the same sets in L as it does in V. Since F(α) builds all the sets in L, this implies that F(α) is a one-to-one correspondence between the class of ordinals and the class of all sets.

http://math.bu.edu/people/aki/9.pdf Ulam

http://math.bu.edu/people/aki/10.pdf Zermelo

http://math.bu.edu/people/aki/11.pdf Levy

http://math.bu.edu/people/aki/12.pdf Gödel (short article)

In his 1940 monograph, based on 1938 lectures, Gödel formulated L via a transfinite recursion that generated L set by set. His incompleteness proof had featured “Gödel numbering”, the encoding of formulas by natural numbers, and his L recursion was a veritable Gödel numbering with ordinals, one that relies on their extent as given beforehand to generate a universe of sets. This approach may have obfuscated the satisfaction aspects of the construction, but on the other hand it did make more evident other aspects: Since there is a direct, definable well-ordering of L, choice functions abound in L, and AC holds there. Also, L was seen to have the important property of absoluteness through the simple operations involved in Gödel’s recursion, one consequence of which is that for any inner model M, the construction of L in the sense of M again leads to the same class L. Decades later many inner models based on first-order definability would be investigated for which absoluteness considerations would be pivotal, and Gödel had formulated the canonical inner model, rather analogous to the algebraic numbers for fields of characteristic zero.

http://math.bu.edu/people/aki/13.pdf Gödel (long article)

http://math.bu.edu/people/aki/14.pdf Cohen

http://math.bu.edu/people/aki/15.pdf Italian article

http://math.bu.edu/people/aki/16.pdf Set theory from Cantor to Cohen

http://math.bu.edu/people/aki/17a.pdf Bernays and Set Theory

http://math.bu.edu/people/aki/18.pdf Suslin's problem

http://math.bu.edu/people/aki/19.pdf Kunen

http://math.bu.edu/people/aki/20.pdf In praise of Replacement

http://math.bu.edu/people/aki/21.pdf Large cardinals with forcing

http://math.bu.edu/people/aki/22.pdf The Mathematical Infinite as a Matter of Method

http://math.bu.edu/people/aki/23.pdf Mathematical Knowledge: Motley and Complexity of Proof

Extra from Zermelo's models and the axiom of limitation of size

The proof of the axiom of global choice in V_κ is more direct than von Neumann's proof. First note that κ (being a von Neumann cardinal) is a well-ordered class of cardinality κ. Since Theorem 2 states that V_κ has cardinality κ, there is a one-to-one correspondence between κ and V_κ. This correspondence produces a well-ordering of V_κ, which implies the axiom of global choice.^[9]

^ Moore 1982, pp. 158–160. Moore argues that the latter was his primary motivation.
^ Moore devotes a chapter to this criticism: "Zermelo and His Critics (1904–1908)", Moore 1982, pp. 85–141.
^ Moore 1982, pp. 158–160. Zermelo 1908, pp. 263–264; English translation: van Heijenoort 1967, p. 202.
^ Cantor 1883, p. ___; English translation: Ewald 1996, p. 891.
^ Cite error: The named reference Godel1940 was invoked but never defined (see the help page).
^ Gödel 1940, p. 27.
^ Gödel 1940, pp.35–38.
^ Bell 1937, pp. 568–569; Hardy and Wright 1938, p. 159 (6th ed., pp. 205–206); Birkhoff and Mac Lane 1941, p. 392, (5th ed., pp. 436–437); Spivak 1967, pp. 369–370 (4th ed., pp. 448–449).
^ The domain of the global choice function consists of the non-empty sets of V_κ; this function uses the well-ordering of V_κ to choose the least element of each set.

Miscellaneous

Shorter examples of wrapping problems in Internet Explorer

In the following sentence, Internet Explorer wraps the subscript between the "g" and "(": x_g(n).

In the following sentence, Internet Explorer wraps the subscript between the "g" and "(": g(n).

I think that it's ridiculous that in such simple expressions, an editor needs to use a nowrap template to avoid the wrapping. Wikipedia should take care of it for us.

I fixed the subsection "Example of Cantor's construction" so that, with one small exception, it wraps the way I want it to in Chrome. Internet Explorer makes many simple wrapping errors such as the ones above.

I can fix it by using the nowrap template on "x_g(n)", but this shouldn't Problems with " " in Internet Explorer and with "nowrap" in Chrome

Thank you David for telling me that I may be seeing a bug. I've done some more experimenting and found a sentence that I can get to wrap properly in either Chrome and Internet Explorer, but not both.

The following sentence which uses " " wraps properly in Chrome, but not in Internet Explorer. For example, by shrinking your window, the 3 occurrences of "g(" can wrap before the "(".

A one-to-one correspondence between T and R is given by the function: g(t) = t if t ∈ T₀, g(t_{2n – 1}) = t_n, and g(t_2n) = a_n.

The next sentence which uses "nowrap" wraps properly in Internet Explorer, but not in Chrome. The 1st occurrence works for Chrome, but Chrome can wrap within the subscripts in the next 2 occurrences.

A one-to-one correspondence between T and R is given by the function: g(t) = t if t ∈ T₀, g(t_{2n – 1}) = t_n, and g(t_2n) = a_n.

So I've just fixed the 1st occurrence so that at least it works in both browsers. I find these examples very interesting. I am specifying the same no-wrap regions in two different ways. Since I'm getting two different behaviors in Chrome or Internet Explorer, it seems unclear whether the bug is in a browser or in Wikipedia code. However, since I can get the proper no-wrapping behavior by using different text the problem can be fixed in Wikipedia. Actually, I could do it myself with we had a browser template so I could write: {{browser | Chrome | … }} {{browser | Internet Explorer | … }} {{browser | default | … }} . Of course, it would be preferable for this to be done by the Wikipedia people who maintain " " and "nowrap". Can my examples be communicated to them? Thanks, SIGN

In the following sentence, by shrinking your window, in the first use of nowrap, you can get the subscript 1 in "a_n₁" or one of the other subscripts to wrap to the next line. In the second use of nowrap, you can get the comma in "n_ν," to wrap to the next line. Is there anywhere we can submit a bug report on nowrap? I switched over to " ".

In his letter introducing the concept of countability, Cantor stated without proof that the set of positive rational numbers is countable, as are sets of the form (a_{n₁, n₂, …, n_ν}) where n₁, n₂, …, n_ν, and ν are positive integers.

In the following sentence, nowrap is necessary because sfrac can cause wrap after "(":

The function can be quite general—for example, a_{n₁, n₂, n₃, n₄, n₅} = (⁠n₁n₂⁠)^⁠1/n₃⁠ + tan(⁠n₄n₅⁠).

In his letter introducing the concept of countability, Cantor stated without proof that the set of positive rational numbers is countable, as are sets of the form (a_{n₁, n₂, …, n_ν}) where n₁, n₂, …, n_ν, and ν are positive integers.

Your GA nomination of Curve-shortening flow (From David Eppstein's talk page: interesting because of "In the news" or "Did you know")

The article Curve-shortening flow you nominated as a good article has passed ; see Talk:Curve-shortening flow for comments about the article. Well done! If the article has not already been on the main page as an "In the news" or "Did you know" item, you can nominate it to appear in Did you know. Message delivered by Legobot, on behalf of Mark viking -- Mark viking (talk) 19:21, 23 April 2016 (UTC)

In the 1890's, in letters to Hilbert, Cantor started calling

I'd like to address the two comments: "rm interpolated sentence which is … unlikely to be true, if there is a genuine disagreement" and "I agree: this sentence being removed seems to express a strange opinion that is contradicted by the well-cited sentences before it."

The analysis of Cantor's proof that shows it to be constructive does contradict the opinions of the mathematicians who regard the proof as non-constructive, but it does not contradict those who regard the proof as constructive. Personally, I know how confusing this can be. Years ago, I read some math history books and popular math books, and learned that Cantor's proof is non-constructive. Later, I came across some books that pointed out that Cantor's methods are constructive. I found the disagreement between these books confusing because a proof cannot be both constructive and non-constructive. So I read Cantor's article and found that his proof is constructive. At first, I couldn't understand how some books could be in error about his proof, but I've found some reasons why this has happened. (If you're interested, I can discuss this on this Talk page.)

So the math literature is confusing, but I've used it to point out an advantage that Wikipedia has over some books. People who know I contribute to Wikipedia have asked me how accurate Wikipedia articles are. I tell them that with so many people reading and correcting them, Wikipedia articles can be more accurate than some books (and the articles provide references to check, which books don't always have). This Wikipedia article is an example of this.

Thanks again to Carl and William. Hopefully, the new sentence handles your concerns. I'm trying to guide readers, who may find the disagreement between mathematicians confusing, to the method that mathematics provides to determine whether Cantor's proof is constructive—namely, the analysis of his article to see if it constructs transcendentals. Hopefully, these readers will then be interested enough to continue reading through the analysis sections.

(For the future if the discussion continues) I do know that all the books and articles I've referenced are reliable sources by Wikipedia standards, but this is a case where some sources are more reliable on this subject than others. (Warning: what comes next contains both facts and personal opinions.) My opinion is that Abraham Fraenkel, a set theorist who carefully studied Cantor's original articles and wrote an excellent biography of Cantor, is definitely more reliable than Birkhoff and MacLane, and Spivak who were busy writing excellent textbooks and probably did not have the time to check Cantor's original article. In fact, I have evidence that Birkhoff and MacLane, and Spivak took their information about "Cantor's proof" from Hardy and Wright's book. In the case of Spivak's Calculus, he gives the same exposition that Hardy and Wright do: Liouville's construction of a transcendental and the non-constructive proof that he attributes to Cantor (pp. 368-370). He states: "Cantor … showed, without exhibiting a single transcendental number, that most numbers are transcendental." In his bibliography (p. 515), he states: "Few books have won so enthusiastic an audience as … An Introduction to the Theory of Numbers (third edition), by G. H. Hardy and E. M. Wright; …." Birkhoff and MacLane also reference Hardy and Wright's book, and I remember they also contrast what they call "Cantor's proof" with Liouville's proof. (I don't have immediate access to their book so I can't give you the page numbers.)

I have also figured out where E. T. Bell and G. H. Hardy probably took their information about "Cantor's proof" from. It seems to me that the mathematicians who say that Cantor's proof is non-constructive are talking about something outside of their area of expertise. Modern set theorists, mathematical logicians, and recursion theorists all seem to think that Cantor's proof is constructive (perhaps because the diagonal argument is used constructively in the first of Gödel's incompleteness theorems and in the Halting problem). There are mathematicians outside these areas who consider Cantor's proof to be constructive. For example, the Wikipedia article quotes Irving Kaplansky, an algebraist. However, after the remarks I quote, he says: "(I owe these remarks to R. M. Robinson.) Robinson was a mathematical logician.

For Wikipedia article (Look at Kristen letter below for changes -- don't forget to ref old Monthly article!)

Cantor's article also contains a proof of the existence of transcendental numbers.^[1] As early as 1930, mathematicians have disagreed on whether Cantor's proof is constructive or non-constructive.^[2] Books as recent as 2014 and 2015 indicate that this disagreement has not been resolved.^[3] A careful study of Cantor's article will determine whether or not his proof is constructive. Cantor's correspondence with Richard Dedekind shows the development of his ideas and reveals that he had a choice between two proofs, one that uses the uncountability of the real numbers and one that does not.

Hi Kristen,

Good news! I just found out that the title of the Wikipedia article I wrote will not be changed because of a lack of consensus.

Good point about enumeration being packed with meaning. I was planning to link the word "enumeration" and found out that the Enumeration article states: "An enumeration is a complete, ordered listing of all the items in a collection. The term is commonly used in mathematics and theoretical computer science (as well as applied computer science) to refer to a listing of all of the elements of a set." So this indicates that it shouldn't be used. The correct mathematical word to use here is "sequence" but as you noted this can be confusing to the reader since we have a sequence of sequences, which is probably why "enumeration" was used. Your suggestion of "list" may be the best alternative even though it's not a term used in mathematics.

Georg Cantor's first set theory article was published in 1874 and contains the first theorems of transfinite set theory, which studies infinite sets and their properties.^[4] One of these theorems is "Cantor's revolutionary discovery" that the set of all real numbers is uncountably, rather than countably, infinite.^[5] This theorem is proved using Cantor's first uncountability proof, which differs from the more familiar proof using his diagonal argument. The title of the article, "On a Property of the Collection of All Real Algebraic Numbers," refers to its first theorem: the set of real algebraic numbers is countable.^[6]

On a Property of the Collection of All Real Algebraic Numbers is Georg Cantor's first set theory article. It was published in 1874 in Crelle's Journal, and it contains the first theorems of transfinite set theory, which studies infinite sets and their properties.^[7] One of these theorems is "Cantor's revolutionary discovery" that the set of all real numbers is uncountably, rather than countably, infinite.^[8] This theorem is proved using Cantor's first uncountability proof, which differs from the more familiar proof using his diagonal argument. The title of the article refers to its first theorem: the set of real algebraic numbers is countable.^[6]

^ Cantor 1874. English translation: Ewald 1996, pp. 840–843.
^ "[Cantor's method is] a method that incidentally, contrary to a widespread interpretation, is fundamentally constructive and not merely existential." (Fraenkel 1930, p. 237; English translation: Gray 1994, p. 823.)
^ "Cantor's proof of the existence of transcendental numbers is not just an existence proof. It can, at least in principle, be used to construct an explicit transcendental number." (Sheppard 2014, p. 131.) "Meanwhile Georg Cantor, in 1874, had produced a revolutionary proof of the existence of transcendental numbers, without actually constructing any." (Stewart 2015, p. 285.)
^ Ferreirós 2007, p. 171.
^ Dauben 1993, p. 4.
^ ^a ^b Ferreirós 2007, p. 177.
^ Cantor 1874; English translation: Ewald 1996, pp. 840–843. Ferreirós 2007, p. 171.
^ Dauben 1993, p. 4.

On a Property of the Collection of All Real Algebraic Numbers is Georg Cantor's first set theory article. It was published in 1874 in Crelle's Journal, and it contains "Cantor's revolutionary discovery" that the set of all real numbers is uncountably, rather than countably, infinite.^[1]. This theorem is proved using Cantor's first uncountability proof, which differs from the more familiar proof using his diagonal argument. The title of the article, "On a Property of the Collection of All Real Algebraic Numbers," refers to its first theorem: the set of real algebraic numbers is countable.^[2]

^ Cantor 1874; English translation: Ewald 1996, pp. 840–843. Dauben 1993, p. 4.
^ Ferreirós 2007, p. 177.

Cantor's article also contains a proof of the existence of transcendental numbers.^[1] As early as 1930, mathematicians have disagreed on whether this proof is constructive or non-constructive.^[2] Books as recent as 2014 and 2015 indicate that this disagreement has not been resolved.^[3] A careful study of Cantor's article will determine whether or not his proof is constructive. Cantor's correspondence with Richard Dedekind shows the development of his ideas and reveals that he had a choice between two proofs, one that uses the uncountability of the real numbers and one that does not.

Historians of mathematics have examined Cantor's article and the circumstances in which it was written. For example, they have discovered that Cantor was advised to leave out his uncountability theorem in the article he submitted; he added it during proofreading.^[4] They have traced this and other facts about the article to the influence of Karl Weierstrass and Leopold Kronecker. Historians have also studied Dedekind's contributions to the article, including his contributions to the theorem on the countability of the real algebraic numbers. In addition, they have looked at the article's legacy—namely, the impact of the uncountability theorem and the concept of countability on mathematics.

^ Cantor 1874. English translation: Ewald 1996, pp. 840–843.
^ "[Cantor's method is] a method that incidentally, contrary to a widespread interpretation, is fundamentally constructive and not merely existential." (Fraenkel 1930, p. 237; English translation: Gray 1994, p. 823.)
^ "Cantor's proof of the existence of transcendental numbers is not just an existence proof. It can, at least in principle, be used to construct an explicit transcendental number." (Sheppard 2014, p. 131.) "Meanwhile Georg Cantor, in 1874, had produced a revolutionary proof of the existence of transcendental numbers, without actually constructing any." (Stewart 2015, p. 285.)
^ Ferreirós 2007, p. 184.

[3] This proof is a simplification of an earlier proof that Cantor that had sent to Richard Dedekind.

[1] Ferreirós 2007, p. 171.

[2] Dauben 1993, p. 4.

[4] Noether & Cavaillès 1937, pp. 14–15. English translation: Ewald 1996, pp. 845–846.

[Dec7letter-5] ^ ^a ^b ^c ^d ^e ^f Cite error: The named reference Dec7letter was invoked but never defined (see the help page).

[6] Noether & Cavaillès 1937, p. 19. English translation: Ewald 1996, p. 849.

[7] Ewald 1996, p. 843.

[8] Noether & Cavaillès 1937, p. 16. English translation: Gray 1994, p. 827.

[9] Kanamori 2009, pp. 48, 58. Bernays' articles are reprinted in Müller 1976, pp. 1–117.

[11] Begin efn #1 ^[1] End efn #1

[14] Begin efn #2 ^[3] End efn #2

[16] Begin efn #3 ^[4]

[10] Outer reference (template): record of Chaplin's birth. Date of his birth. End of outer reference

[12] Inner ref: Robinson, p. xxiv. End of inner ref

[13] Outer reference (template): record of Chaplin's birth. Date of his birth.^[2] End of outer reference

[15] Inner ref: Robinson, p. xxiv. End of inner ref

[17] Inner ref #3

[37] In his relative consistency proof, Gödel defined a function $F$ on the ordinals such that $F(\alpha )$ is a set built by applying a set operation to sets $F(\beta )$ where $\beta <\alpha .$ The constructible universe is the image of $F.$ Gödel's set operations come from the axiom of pairing, the class existence axioms restricted to sets, and taking the union of the preceding classes. Cohen modified Gödel's function to create structured, but undefined symbols that would ultimately be the sets needed to prove an independence theorem. For the independence of the continuum hypothesis, he introduced the symbols $a_{\delta }$ where $\delta <\aleph _{\tau }$ and $\tau \geq 2.$ Later, each $a_{\delta }$ would become a different subset of $\omega$ so the continuum hypothesis fails in the model. Cohen started by defining the values $F_{\alpha }=\alpha$ for $\alpha \leq \omega .$ Next, $F_{\alpha }$ successively enumerates $a_{\delta }$ for $\delta <\aleph _{\tau },\{a_{\delta },a_{\delta '}\}$ for $\delta <\delta ',$ and $(a_{\delta },a_{\delta '})$ for $\delta <\delta '.$ Let $F_{3\aleph _{\tau }}=\{(a_{\delta },a_{\delta '}):\delta <\delta '\}.$ Finally, $F_{\alpha }$ is defined for $\alpha >3\aleph _{\tau }$ by applying Gödel's set-building operations to $F(\beta )$ where $\beta <\alpha .$ (Cohen 1963, pp. 1144–1145.)

[38] Kanamori 2009, p. 65: "Forcing itself went a considerable distance in downgrading any formal theory of classes because of the added encumbrance of having to specify the classes of generic extensions."

[40] In the 1960s, this conservative extension theorem was proved independently by Paul Cohen, Saul Kripke, and Robert Solovay. In his 1966 book, Cohen mentioned this theorem and stated that its proof requires forcing. It was also proved independently by Ronald Jensen and Ulrich Felgner, who published his proof in 1971. (Ferreirós 2007, pp. 381–382; Cohen 1966, p. 77; Felgner 1971.)

[18] Dauben 1979, p. 66–70. Gray 1994 (p. 828) also gives an example: "… [Cantor's] theorems only deal with sequences that are ordered by a 'law.'" Gray states that "Cantor may have inserted this restriction to avoid problems with Kronecker." However, Ferreirós 2007 (p. 263) corrects Gray: "Like all other authors in the early period of the history of sets, he [Cantor] tended to think of them [sets] as given by a concept or a law, indeed he emphasized this aspect more than other contemporaries." Ferreirós provides examples from Cantor's articles dating from 1872 to 1883.

[19] There are $2^{\aleph _{0}}$ points on a line segment, $(2^{\aleph _{0}})^{2}$ points on a plane, and $(2^{\aleph _{0}})^{n}$ points in $n$ -dimensional space. Since $(2^{\aleph _{0}})^{n}=2^{\aleph _{0}\times n}=2^{\aleph _{0}},$ there are the same number of points on a line segment, on a plane, and in $n$ -dimensional space.

[standard-20] Quantities and Units - Part 2: Mathematical signs and symbols to be used in the natural sciences and technology, page 15. ISO 80000-2 (ISO/IEC 2009-12-01)

[21] This definition of image is used in the proof that NBG's axiom of replacement implies NBG's axiom of separation. See NBG's axiom of separation.

[22] This definition of image is used in the proof that NBG's axiom of replacement implies NBG's axiom of separation. See NBG's axiom of separation.

[23] Gödel 1940, p. 16; Jech 2003, p. 11; Cunningham 2016, p. 57. Gödel gives definitions 1 and 2 (he uses "over X " instead of "on X " in definition 2). Jech gives all three definitions; Cunningham gives definitions 1 and 3.

[cantor1874-24] Cantor, Georg (1874), "Ueber eine Eigenschaft des Inbegriffes aller reellen algebraischen Zahlen", J. Reine Angew. Math., 77: 258–262, doi:10.1515/crll.1874.77.258

[25] Johnson, Philip (1972), A History of Set Theory, Prindle, Weber & Schmidt, ISBN 0-87150-154-6

[26] Bolzano, Bernard (1975), Berg, Jan (ed.), Einleitung zur Größenlehre und erste Begriffe der allgemeinen Größenlehre, Bernard-Bolzano-Gesamtausgabe, edited by Eduard Winter et al., vol. Vol. II, A, 7, Stuttgart, Bad Cannstatt: Friedrich Frommann Verlag, p. 152, ISBN 3-7728-0466-7 {{citation}}: |volume= has extra text (help)

[27] Dauben, Joseph (1979), Georg Cantor: His Mathematics and Philosophy of the Infinite, Harvard University Press, pp. 30–54, ISBN 0-674-34871-0.

[28] BIO of Zermelo

[29] In Russell's set theory, every Cantorian multiplicity is a set so Russell obtains what he calls Cantor's paradox. Purkert, Walter (1989), "Cantor's Views on the Foundations of Mathematics", in Rowe, David E.; McCleary, John (eds.) (eds.), The History of Modern Mathematics, Volume 1, Academic Press, p. 56, ISBN 0-12-599662-4 {{citation}}: |editor-first2= has generic name (help).

[30] Zermelo, Ernst (1930), "Über Grenzzahlen und Mengenbereiche: neue Untersuchungen über die Grundlagen der Mengenlehre" (PDF), Fundamenta Mathematicae, 16: 29–47. English translation: Ewald, William B. (1996), "On boundary numbers and domains of sets: new investigations in the foundations of set theory", From Immanuel Kant to David Hilbert: A Source Book in the Foundations of Mathematics, Oxford University Press, pp. 1208–1233, ISBN 978-0-19-853271-2.

[31] Von_Neumann–Bernays–Gödel_set_theory#History.

[32] Ferreirós, José (2007), Labyrinth of Thought: A History of Set Theory and Its Role in Mathematical Thought (2nd revised ed.), Birkhäuser, pp. 381–382, ISBN 978-3-7643-8349-7.

[33] Blair, Charles E. The Baire category theorem implies the principle of dependent choices. Bull. Acad. Polon. Sci. Sér. Sci. Math. Astronom. Phys. 25 (1977), no. 10, 933–934.

[34] Moore 1982, p. 325. Moore's table states that "Principle of Dependent Choices" $\rightarrow$ "Löwenheim–Skolem theorem"—that is, ${\mathsf {DC}}$ implies the Löwenheim–Skolem theorem. The converse is proved in Boolos and Jeffrey 1989, pp. 155–156.

[35] Blair, Charles E. The Baire category theorem implies the principle of dependent choices. Bull. Acad. Polon. Sci. Sér. Sci. Math. Astronom. Phys. 25 (1977), no. 10, 933–934.

[36] Kanamori 2009, p. 57.

[39] Cohen 1966, pp. 107–147. Cohen also gave a detailed proof of Gödel's relative consistency theorems using ZF (Cohen 1966, pp. 85–99).

[43] Kanamori 2009, p. 65: "Forcing itself went a considerable distance in downgrading any formal theory of classes because of the added encumbrance of having to specify the classes of generic extensions."

[45] In the 1960s, this conservative extension theorem was proved independently by Paul Cohen, Saul Kripke, and Robert Solovay. In his 1966 book, Cohen mentioned this theorem and stated that its proof requires forcing. It was also proved independently by Ronald Jensen and Ulrich Felgner, who published his proof in 1971. (Ferreirós 2007, pp. 381–382; Cohen 1966, p. 77; Felgner 1971.)

[41] Kanamori 2009, p. 57.

[42] Cohen 1963.

[44] Cohen 1966, pp. 107–147. Cohen also gave a detailed proof of Gödel's relative consistency theorems using ZF (Cohen 1966, pp. 85–99).

[46] Moore 1982, pp. 158–160. Moore argues that the latter was his primary motivation.

[47] Moore devotes a chapter to this criticism: "Zermelo and His Critics (1904–1908)", Moore 1982, pp. 85–141.

[48] Moore 1982, pp. 158–160. Zermelo 1908, pp. 263–264; English translation: van Heijenoort 1967, p. 202.

[49] Cantor 1883, p. ___; English translation: Ewald 1996, p. 891.

[Godel1940-50] Cite error: The named reference Godel1940 was invoked but never defined (see the help page).

[51] Gödel 1940, p. 27.

[52] Gödel 1940, pp.35–38.

[53] Bell 1937, pp. 568–569; Hardy and Wright 1938, p. 159 (6th ed., pp. 205–206); Birkhoff and Mac Lane 1941, p. 392, (5th ed., pp. 436–437); Spivak 1967, pp. 369–370 (4th ed., pp. 448–449).

[54] The domain of the global choice function consists of the non-empty sets of V_κ; this function uses the well-ordering of V_κ to choose the least element of each set.

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Misc

Cantor at FAC (part 2)

Locke, An Essay concerning Human Understanding

Featured article info

From: Editing Help:Wikipedia: The Missing Manual/Collaborating with other editors

Going for the gold: Better and best article candidates

To work on: (Cantor's 1883 article): User:RJGray/Sandboxcantor1

Free ref for Mathematische Annalen. Backup if use of current one goes bad.

Michael Hardy

Iry-Hor

Ferreiros

Michael Hardy: Addition to article & Featured article nomination

ADD TO: "The development of Cantor's ideas"

Notes

References

Stuff for monthly article

Other to do

Notes

References

Jochen (#1: Reference within efn)

Notes

References

Need some advice

Reply

Notes

References

Havil irrational book

Fixed broken links and rewrote "In set theory"

History (From "Set Theory")

Subcompact cardinal (lead)

See also

References

Equivalent statements

Jochen

Rank-into-rank cardinals

Gödel/Cohen

Add to ZFC

Error in Zermelo set theory: The aim of Zermelo's paper

Locke, Descartes, Spinoza, Leibnitz quotes

experiment

Possible note for Cantor's first set theory article?

Gödel's work with proper classes

Talk for Cantor's first set theory article: Connections between books that claim Cantor's proof is non-constructive

Future

Talk: Is von Neumann's 1923 axiom weaker than or equivalent to the axiom of limitation of size?

Wikipedia can be more accurate than some books

Gödel's L and the axiom of limitation of size

Extra from Zermelo's models and the axiom of limitation of size

Miscellaneous