Cantor's diagonal argument

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Cantor's diagonal argument is a proof 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.)

Contrary to what many mathematicians believe, the diagonal argument was not Cantor's first proof of the uncountability of the real numbers, but was published three years later. His original argument did not mention decimal expansions, nor any other numeral system.

Since this technique was first used, similar proof constructions have been used many times in a wide range of proofs. These are also known as diagonal arguments by analogy with the argument used in this proof.

Cantor's original proof shows that the interval [0,1] is not countably infinite.

The proof by contradiction proceeds as follows:

(1) Assume 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 in [0,1] by considering the n^th digit after the decimal point of the decimal expansion of r_n. In the case of numbers with two decimal expansions, like 0.499 ... = 0.500 ..., we chose the one ending in nines.

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 in [0,1].

(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.

A generalized form of the diagonal argument was used by Cantor to prove Cantor's theorem: 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:

Let f be any one-to-one function from S to P(S). It suffices to prove f cannot be surjective. That means that some member of P(S), i.e., some subset of S, is not in the image of f. That set is

If T is in the range of f, then for some t in S we have T = f(t). Either t is in T or not. If t is in T, then t is in f(t), but, by definition of T, that implies t is not in T. On the other hand, if t is not in T, then t is not in f(t), and by definition of T, that implies t is in T. Either way, we have a contradition.

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.

Analogues of the diagonal argument are widely used in mathematics to prove the existence or nonexistence of certain objects. For example, the conventional proof of the unsolvability of the halting problem is essentially a diagonal argument.

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.