Cantor's diagonal argument

Cantor's diagonal argument is a logical argument devised by Georg Cantor to demonstrate that the real numbers are not countably infinite. (It is also called the diagonalization argument or the diagonal slash argument.) It does this by showing that the interval (0,1), that is, the real numbers larger than 0 and smaller than 1, is not countably infinite. It proceeds as follows:

• (1) Assume for a contradiction that the interval (0,1) is countably infinite.
• (2) We may then enumerate the numbers in this interval as a sequence, { r₁, r₂, r₃, ... }
• (3) We shall now construct a real number x between 0 and 1 by considering the n^th digit after the decimal point of the decimal expansion of r_n. Assume, for example, that the decimal expansions of the beginning of the sequence are as follows.

      r1 = 0 . 0 1 0 5 1 1 0 ... 
      r2 = 0 . 4 1 3 2 0 4 3 ...
      r3 = 0 . 8 2 4 5 0 2 6 ... 
      r4 = 0 . 2 3 3 0 1 2 6 ...
      r5 = 0 . 4 1 0 7 2 4 6 ... 
      r6 = 0 . 9 9 3 7 8 3 8 ...
      r7 = 0 . 0 1 0 5 1 3 0 ... 
      ...

The digits we will consider are indicated in bold. From these digits we define the digits of x as follows.

• • if the n^th digit of r_n is 0 then the n^th digit of x is 1
  • if the n^th digit of r_n is not 0 then the n^th digit of x is 0

For the example above this will result in the following decimal expansion.

       x = 0 . 1 0 0 1 0 0 1 ...

The number x is clearly a real number between 0 and 1.

• (4) However, it differs in the n^th decimal place from r_n, so x is not in the set { r₁, r₂, r₃, ... }.
• (5) This set is therefore not an enumeration of all the reals in the interval (0,1).
• (6) This contradicts with (2), so the assumption (1) that the interval (0,1) is countably infinite must be false.

Note: Strictly speaking, this argument only shows that the number of decimal expansions of real numbers between 0 and 1 is not countably infinite. But since there are expansions such as 0.01999... and 0.02000... that represent the same real number, this does not immediately imply that the corresponding set of real numbers is also not countably infinite. This can be remedied by disallowing the decimal expansions that end with an infinite series of 9's. In that case it can be shown that for every real number there is a unique corresponding decimal expansion. It is easy to see that the proof then still works because the number x contains only 1's and 0's in its decimal expansion.

The diagonal argument is an example of reductio ad absurdum because it proves a certain proposition (the interval (0,1) is not countably infinite) by showing that the assumption of its negation leads to a contradiction.

A generalized form of the diagonal argument was used by Cantor to show that for every set S the power set of S, i.e., the set of all subsets of S (here written as P(S)), is larger than S itself. This proof proceeds as follows:

• (1) Assume that P(S) is not larger than S.
• (2) Then there is a function f from S to P(S) that is surjective, i.e., for every V in P(S) there is an s in S such that f(s) = V.
• (3) We shall now construct a set T that is in P(S) as follows.

T = { s in S | s is not in f(s) }

• (4) The set T is not in the range of f, i.e., there is no s in S such that f(s) = T. This can be shown as follows. It will either hold that s is in f(s) or not.
  • If s is in f(s) then it follows by definition of T that s is not in T, so T is not equal to f(s).
  • If s is not in f(s) then it follows by definition of T that s is in T, so also then T is not equal to f(s).

Since in both cases f(s) is not equal to T it follows that f(s) is never equal to T.

• (5) The function f is therefore not surjective.
• (6) This contradicts with (2), so the assumption (1) that P(S) is not larger than S must be false.

Note the similarity between the construction of T and the set in Russell's paradox. Its result can be used to show that the notion of the set of all sets is an inconsistent notion in normal set theory; if S would be the set of all sets then P(S) would at the same time be bigger than S and a subset of S.

The diagonal argument shows that the set of real numbers is "bigger" than the set of integers. Therefore, we can ask if there is a set whose cardinality is "between" that of the integers and that of the reals. This question leads to the famous Continuum hypothesis. Similarly, the question of whether there exists a set whose cardinality is between s and P(s) for some s, leads to the Generalized continuum hypothesis.