CANTOR'S DIAGONAL ARGUMENT

'Cantor's diagonal argument', also called the 'diagonalisation 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, the first of Gödel's incompleteness theorems, and Turing's answer to the Entscheidungsproblem.

Contents

An uncountable set

Real numbers

General sets

External links

An uncountable set

Cantor's original proof considers an infinite sequence of the form (''x''₁, ''x''₂, ''x''₃, ...) where each element ''x''_i is either 0 or 1.
Consider any infinite listing of some of these sequences. We might have for instance:
:''s''₁ = (0, 0, 0, 0, 0, 0, 0, ...)
:''s''₂ = (1, 1, 1, 1, 1, 1, 1, ...)
:''s''₃ = (0, 1, 0, 1, 0, 1, 0, ...)
:''s''₄ = (1, 0, 1, 0, 1, 0, 1, ...)
:''s''₅ = (1, 1, 0, 1, 0, 1, 1, ...)
:''s''₆ = (0, 0, 1, 1, 0, 1, 1, ...)
:''s''₇ = (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 ''s''₀ 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:
:''s''₁ = ('0', 0, 0, 0, 0, 0, 0, ...)
:''s''₂ = (1, '1', 1, 1, 1, 1, 1, ...)
:''s''₃ = (0, 1, '0', 1, 0, 1, 0, ...)
:''s''₄ = (1, 0, 1, '0', 1, 0, 1, ...)
:''s''₅ = (1, 1, 0, 1, '0', 1, 1, ...)
:''s''₆ = (0, 0, 1, 1, 0, '1', 1, ...)
:''s''₇ = (1, 0, 0, 0, 1, 0, '0', ...)
:...
:''s''₀ = ('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". Note that the highlighted element in ''s''₀ is in every case different from the highlighted element in the table above it.)
Therefore it may be seen that this new sequence ''s''₀ 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 ''s''₀, 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 ''s''₁, ''s''₂, ''s''₃, ... Otherwise, it would be possible by the above process to construct a sequence ''s''₀ 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 ''s''₀, which is just such a sequence. But since ''s''₀ does not appear anywhere on the list, ''T'' cannot contain ''s''₀.
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.

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.

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
:$T=\{,sin\; S:\; s$
otin f(s),}.
If ''T'' is in the image 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.
This result implies that the notion of the set of all sets is an inconsistent notion. 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''.
For a more complete account of this proof, see Cantor's theorem.
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 infinite ''S'' leads to the generalized continuum hypothesis.
Russell's Paradox has shown us that naive set theory, based on an unrestricted comprehension scheme, is contradictory. Note that there is a similarity between the construction of ''T'' and the set in Russell's paradox. Therefore, depending on how we modify the axiom scheme of comprehension in order to avoid Russell's paradox, arguments such as the non-existence of a set of all sets may or may not remain valid.
So, for example, the above proof fails for W. V. Quine's "New Foundations" set theory (NF). In NF, the naive axiom scheme of comprehension is modified to avoid the paradoxes by introducing a kind of "local" type theory. In this axiom scheme, $\{s\; in\; S:\; s$
otin f(s),} is 'not' a set - ie, does not satisfy the axiom scheme. On the other hand, we might try to create a modified diagonal argument by noticing that $\{s\; in\; S:\; s$
otin f({s}),} 'is' a set in NF. In which case, if '''P'''₁(''S'') is the set of one-element subsets of S and f is a proposed bijection from '''P'''₁(''S'') to '''P'''(''S''), one is able to use reductio to prove that |'''P'''₁(''S'')| < |'''P'''(''S'')|.
The proof follows by the fact that if ''f'' were indeed a map 'onto' '''P'''(''S'')), then we could find $r\; in\; S\; |\; f(\; \{\; r\; \}\; )$ coincides with the modified diagonal set, above. We would conclude that if $r$
otin f({r}), then $r\; in\; f(\{r\})$ and visa-versa.
It is 'not' possible to put '''P'''₁(''S'') in 1-1 relation with ''S'', as the two have different types, and so any function so defined would violate the typing rules for the comprehension scheme.
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.

External links

★ Original German text of the 1891 proof, with English translation