In set theory, Cantor's diagonal argument, also called the diagonalisation argument, the diagonal slash argument, the anti-diagonal argument, or the diagonal method, was published in 1891 by Georg Cantor as a mathematical proof that there are infinite sets which cannot be put into one-to-one correspondence with the infinite set of natural numbers.^{[1]}^{[2]}^{:20-}^{[3]} 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, which appeared in 1874.^{[4]}^{[5]} However, it demonstrates a general technique that has since been used in a wide range of proofs,^{[6]} including the first of Gödel's incompleteness theorems^{[2]} and Turing's answer to the Entscheidungsproblem. Diagonalization arguments are often also the source of contradictions like Russell's paradox^{[7]}^{[8]} and Richard's paradox.^{[2]}^{:27}
In his 1891 article, Cantor considered the set T of all infinite sequences of binary digits (i.e. each digit is zero or one). He begins with a constructive proof of the following theorem:
The proof starts with an enumeration of elements from T, for example:
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, | ...) |
... |
Next, a sequence s is constructed by choosing the 1st digit as complementary to the 1st digit of s_{1} (swapping 0s for 1s and vice versa), the 2nd digit as complementary to the 2nd digit of s_{2}, the 3rd digit as complementary to the 3rd digit of s_{3}, and generally for every n, the n^{th} digit as complementary to the n^{th} digit of s_{n}. For the example above, this yields:
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 | = | (1, | 0, | 1, | 1, | 1, | 0, | 1, | ...) |
By construction, s differs from each s_{n}, since their n^{th} digits differ (highlighted in the example). Hence, s cannot occur in the enumeration.
Based on this theorem, Cantor then uses a proof by contradiction to show that:
The proof starts by assuming that T is countable. Then all its elements can be written as an enumeration s_{1}, s_{2}, ... , s_{n}, ... . Applying the previous theorem to this enumeration produces a sequence s not belonging to the enumeration. However, this contradicts s being an element of T and therefore belonging to the enumeration. This contradiction implies that the original assumption is false. Therefore, T is uncountable.
The uncountability of the real numbers was already established by Cantor's first uncountability proof, but it also follows from the above result. To prove this, an injection will be constructed from the set T of infinite binary strings to the set R of real numbers. Since T is uncountable, the image of this function, which is a subset of R, is uncountable. Therefore, R is uncountable. Also, by using a method of construction devised by Cantor, a bijection will be constructed between T and R. Therefore, T and R have the same cardinality, which is called the "cardinality of the continuum" and is usually denoted by or .
An injection from T to R is given by mapping strings in T to decimals, such as mapping t = 0111... to the decimal 0.0111.... This function, defined by , is an injection because it maps different strings to different numbers.
Constructing a bijection between T and R is slightly more complicated. Instead of mapping 0111... to the decimal 0.0111..., it can be mapped to the base b number: 0.0111..._{b}. This leads to the family of functions: . The functions are injections, except for . This function will be modified to produce a bijection between T and R.
Construction of a bijection between T and R |
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This construction uses a method devised by Cantor that was published in 1878. He used it to construct a bijection between the closed interval [0, 1] and the irrationals in the open interval (0, 1). He first removed a countably infinite subset from each of these sets so that there is a bijection between the remaining uncountable sets. Since there is a bijection between the countably infinite subsets that have been removed, combining the two bijections produces a bijection between the original sets.^{[9]} Cantor's method can be used to modify the function to produce a bijection from T to (0, 1). Because some numbers have two binary expansions, is not even injective. For example, 0.1000..._{2} = 1/2 and 0.0111..._{2} = 1/2, so both 1000... and 0111... map to the same number, 1/2. To modify , observe that it is a bijection except for a countably infinite subset of (0, 1) and a countably infinite subset of T. It is not a bijection for the numbers in (0, 1) that have two binary expansions. These are called dyadic numbers and have the form where m is an odd integer and n is a natural number. Put these numbers in the sequence: r = (1/2, 1/4, 3/4, 1/8, 3/8, 5/8, 7/8, ...). Also, is not a bijection to (0, 1) for the strings in T appearing after the binary point in the binary expansions of 0, 1, and the numbers in sequence r. Put these eventually-constant strings in the sequence: s = (000..., 111..., 1000..., 0111..., 01000..., 00111..., 11000..., 10111..., ...). Define the bijection g(t) from T to (0, 1): If t is the n^{th} string in sequence s, let g(t) be the n^{th} number in sequence r; otherwise, g(t) = 0.t_{2}. To construct a bijection from T to R, start with the tangent function tan(x), which is a bijection from (-?/2, ?/2) to R (see the figure shown on the right). Next observe that the linear function h(x) = is a bijection from (0, 1) to (-?/2, ?/2) (see the figure shown on the left). The composite function tan(h(x)) = is a bijection from (0, 1) to R. Composing this function with g(t) produces the function tan(h(g(t))) = , which is a bijection from T to R. |
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--that is, the set of all subsets of S (here written as P(S))--cannot be in bijection with S itself. This proof proceeds as follows:
Let f be any function from S to P(S). It suffices to prove f cannot be surjective. That means that some member T of P(S), i.e. some subset of S, is not in the image of f. As a candidate consider the set:
For every s in S, either s is in T or not. If s is in T, then by definition of T, s is not in f(s), so T is not equal to f(s). On the other hand, if s is not in T, then by definition of T, s is in f(s), so again T is not equal to f(s); cf. picture. For a more complete account of this proof, see Cantor's theorem.
Cantor defines an order relation of cardinalities, e.g. and , in terms of the existence of injections between the underlying sets, and . The argument in the previous paragraph then proved that sets such as are uncountable, which is to say , and we can embed the naturals into the function space, so that we have that . In the context of classical mathematics, this exhausts the possibilities and the diagonal argument can thus be used to establish that, for example, although both sets under consideration are infinite, there are actually more infinite sequences of ones and zeros than there are natural numbers. This result then also implies that the notion of the set of all sets is inconsistent: If S were the set of all sets, then P(S) would at the same time be bigger than S and a subset of S. Assuming the law of excluded middle, every subcountable set (a property in terms surjections) is also already countable.
In Constructive mathematics, it is harder or impossible to order ordinals and also cardinals. For example, the Schröder-Bernstein theorem requires the law of excluded middle. The ordering on the reals (the standard ordering extending that of the rational numbers) is also not necessarily decidable. Neither are most properties of interesting classes of functions decidable, by Rice's theorem, i.e. the right set of counting numbers for the subcountable sets may not be recursive. In a set theory, theories of mathematics are modeled. For example, in set theory, "the" set of real numbers is identified as a set that fulfills some axioms of real numbers. Various models have been studied, such as the Dedekind reals or the Cauchy reals. Weaker axioms mean less constraints and so allow for a richer class of models. Consequently, in an otherwise constructive context (law of excluded middle not taken as axiom), it is consistent to adopt non-classical axioms that contradict consequences of the law of excluded middle. For example, the uncountable space of functions may be asserted to be subcountable, a notion of size orthogonal to the statement .^{[10]} In such a context, the subcountability of the real numbers is possible, intuitively making for a thinner set of numbers than in other models.
Motivated by the insight that the set of real numbers is "bigger" than the set of natural numbers, one is led to 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 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.
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. Also, diagonalization was originally used to show the existence of arbitrarily hard complexity classes and played a key role in early attempts to prove P does not equal NP.
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,
is not a set -- i.e., does not satisfy the axiom scheme. On the other hand, we might try to create a modified diagonal argument by noticing that
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 proof by contradiction 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, such that f({r}) coincides with the modified diagonal set, above. We would conclude that if r is not in f({r}), then r is in f({r}) and vice versa.
It is not possible to put P_{1}(S) in a one-to-one relation with S, as the two have different types, and so any function so defined would violate the typing rules for the comprehension scheme.