Cantor's diagonal argument

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Cantor's diagonal argument, also called the diagonalization argument, the diagonal slash argument or the diagonal method, was published in 1891 by Georg Cantor as a proof that there are infinite sets which cannot be put into one-to-one correspondence with the infinite set of natural numbers. Such sets are now known as uncountable sets, and the size of infinite sets is now treated by the theory of cardinal numbers which Cantor began.

The diagonal argument was not Cantor's first proof of the uncountability of the real numbers; it was actually published three years after his first proof, which appears in 1874. However, it demonstrates a powerful and general technique, which has since been reused many times in a wide range of proofs, also known as diagonal arguments by analogy with the argument used in this proof. The most famous examples are perhaps Russell's paradox, Gödel's first incompleteness theorem, and Turing's answer to the Entscheidungsproblem.

1 An uncountable set
- 1.1 Real numbers
2 General sets
3 External links

[edit] An uncountable set

Cantor's original proof considers infinite sequences of elements of the form (x₁, x₂, x₃, ...) where each element x_i is either 0 or 1.

Consider any infinite listing of some of these sequences. We might have for instance:

s₁ = (0, 0, 0, 0, 0, 0, 0, ...)

s₂ = (1, 1, 1, 1, 1, 1, 1, ...)

s₃ = (0, 1, 0, 1, 0, 1, 0, ...)

s₄ = (1, 0, 1, 0, 1, 0, 1, ...)

s₅ = (1, 1, 0, 1, 0, 1, 1, ...)

s₆ = (0, 0, 1, 1, 0, 1, 1, ...)

s₇ = (1, 0, 0, 0, 1, 0, 0, ...)

...

And in general we shall write

s_n = (s_n,1, s_n,2, s_n,3, s_n,4, ...)

that is to say, s_n,m is the m^th element of the n^th sequence on the list.

It is possible to build a sequence of elements s₀ in such a way that its first element is different from the first element of the first sequence in the list, its second element is different from the second element of the second sequence in the list, and, in general, its n^th element is different from the n^th element of the n^th sequence in the list. That is to say, s_0,m will be 0 if s_m,m is 1, and s_0,m will be 1 if s_m,m is 0. For instance:

s₁ = (0, 0, 0, 0, 0, 0, 0, ...)

s₂ = (1, 1, 1, 1, 1, 1, 1, ...)

s₃ = (0, 1, 0, 1, 0, 1, 0, ...)

s₄ = (1, 0, 1, 0, 1, 0, 1, ...)

s₅ = (1, 1, 0, 1, 0, 1, 1, ...)

s₆ = (0, 0, 1, 1, 0, 1, 1, ...)

s₇ = (1, 0, 0, 0, 1, 0, 0, ...)

...

s₀ = (1, 0, 1, 1, 1, 0, 1, ...)

(The elements s_1,1, s_2,2, s_3,3, and so on, are here highlighted, showing the origin of the name "diagonal argument".)

And yet it may be seen that this new sequence s₀ is distinct from all the sequences in the list. This follows from the fact that if it were identical to, say, the 10th sequence in the list, then we would have s_0,10 = s_10,10. In general, if it appeared as the n^th sequence on the list, we would have s_0,n = s_n,n, which, due to the construction of s₀, is impossible.

From this it follows that the set T, consisting of all infinite sequences of zeros and ones, cannot be put into a list s₁, s₂, s₃, ... Otherwise, it would be possible by the above process to construct a sequence s₀ which would both be in T (because it is a sequence of 0's and 1's which is by the definition of T in T) and at the same time not in T (because we can deliberately construct it not to be in the list). T, containing all such sequences, must contain s₀, which is just such a sequence. But since s₀ does not appear anywhere on the list, T cannot contain s₀.

Therefore T cannot be placed in one-to-one correspondence with the natural numbers. In other words, it is uncountable.

The interpretation of Cantor's result will depend upon one's view of mathematics, and more specifically on how one thinks of mathematical functions. In the context of classical mathematics, functions need not be computable, and hence the diagonal argument establishes that, there are more infinite sequences of ones and zeros than there are natural numbers. To those constructivists who countenance only computable functions, Cantor's proof (merely) shows that there is no recursively enumerable set of indices (for example, Gödel numbers) for the programs computing them.

[edit] Real numbers

The uncountability of the real numbers was already established by Cantor's first uncountability proof, but it also follows from this result. It can be shown that the set T can be placed into one-to-one correspondence with the real numbers, that is, it has the cardinality of the continuum. As T is uncountable, it follows that the real numbers must also be uncountable.

[edit] General sets

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

$T=\{\,s\in S: s\not\in f(s)\,\}.$

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 contradiction.

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 above proof fails for W. V. Quine's "New Foundations" set theory, which has a different version of the axiom of comprehension in which $\{s \in S: s\not\in f(s)\,\}$ cannot in general be shown to exist. The set $\{s \in S: s\not\in f(\{s\})\,\}$ (where P₁(S) is the set of one-element subsets of S and f is supposed to be a bijection from P₁(S) to P(S)) can be shown to exist in New Foundations, so the theorem one is able to prove is that |P₁(S)| < |P(S)|: if $f ({r})$ were equal to the set above (which must be true for some $r \in S$ if f is a map onto P(S)), then $r \in f(\{r\})$ would imply and be implied by its own negation.

For a more concrete account of this proof that is possibly easier to understand see Cantor's theorem.

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.