Cantor's diagonal argument (among various similar names[1]) is a mathematical proof that there are infinite sets which cannot be put into one-to-one correspondence with the infinite set of natural numbersinformally, that there are sets which in some sense contain more elements than there are positive integers. Such sets are now called uncountable sets, and the size of infinite sets is treated by the theory of cardinal numbers, which Cantor began.
Georg Cantor published this proof in 1891,[2] [3] [4] but it was not his first proof of the uncountability of the real numbers, which appeared in 1874.[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[3] 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.[3]
Cantor considered the set T of all infinite sequences of binary digits (i.e. each digit is zero or one).[9] He begins with a constructive proof of the following lemma:
If s1, s2, ..., sn, ... is any enumeration of elements from T,[10] then an element s of T can be constructed that doesn't correspond to any sn in the enumeration.The proof starts with an enumeration of elements from T, for example
| 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, | ...) |
| ... |
| 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, | ...) | |
| ... | ||||||||||
| s | = | (1, | 0, | 1, | 1, | 1, | 0, | 1, | ...) |
Based on this lemma, Cantor then uses a proof by contradiction to show that:
The set T is uncountable.The proof starts by assuming that T is countable.Then all its elements can be written in an enumeration s1, s2, ..., sn, ... .Applying the previous lemma to this enumeration produces a sequence s that is a member of T, but is not in the enumeration. However, if T is enumerated, then every member of T, including this s, is in the enumeration. This contradiction implies that the original assumption is false. Therefore, T is uncountable.[2]
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
ak{c}
| \aleph0 | |
| 2 |
An injection from T to R is given by mapping binary strings in T to decimal fractions, 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.[11]
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.
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:
T = .
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.
With equality defined as the existence of a bijection between their underlying sets, Cantor also defines binary predicate of cardinalities
|S|
|T|
S
T
\le
|{N}|\le|2N|
2N
{N}\to\{0,1\}
|{N}|<|2N|
<
|S|<|{lP}(S)|
\neg(|{lP}(S)|\le|S|)
S
Assuming the law of excluded middle, characteristic functions surject onto powersets, and then
|2S|=|{lP}(S)|
2N
{N}
{N}
{N}
{N}
\le
S
{lP}(S)
S
S
Also in constructive mathematics, there is no surjection from the full domain
{N}
{N}N
{lP}({N})
<
{N}<2N
S<{lP}(S)
\neg({lP}(S)\leS)
2N\le{N}N
2S\le{lP}(S)
S\leS
It is however harder or impossible to order ordinals and also cardinals, constructively. For example, the Schröder–Bernstein theorem requires the law of excluded middle.[13] In fact, the standard ordering on the reals, extending the ordering of the rational numbers, is not necessarily decidable either. Neither are most properties of interesting classes of functions decidable, by Rice's theorem, i.e. the set of counting numbers for the subcountable sets may not be recursive and can thus fail to be countable. The elaborate collection of subsets of a set is constructively not exchangeable with the collection of its characteristic functions. In an otherwise constructive context (in which the law of excluded middle is not taken as axiom), it is consistent to adopt non-classical axioms that contradict consequences of the law of excluded middle. Uncountable sets such as
2N
{N}N
2N
{N}N
{N}
In a set theory, theories of mathematics are modeled. Weaker logical axioms mean fewer constraints and so allow for a richer class of models. A set may be identified as a model of the field of real numbers when it fulfills some axioms of real numbers or a constructive rephrasing thereof. Various models have been studied, such as the Cauchy reals or the Dedekind reals, among others. The former relate to quotients of sequences while the later are well-behaved cuts taken from a powerset, if they exist. In the presence of excluded middle, those are all isomorphic and uncountable. Otherwise, variants of the Dedekind reals can be countable[17] or inject into the naturals, but not jointly. When assuming countable choice, constructive Cauchy reals even without an explicit modulus of convergence are then Cauchy-complete[18] and Dedekind reals simplify so as to become isomorphic to them. Indeed, here choice also aids diagonal constructions and when assuming it, Cauchy-complete models of the reals are uncountable.
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 set theory that includes 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 P1(S) is the set of one-element subsets of S and f is a proposed bijection from P1(S) to P(S), one is able to use proof by contradiction to prove that |P1(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 coincides with the modified diagonal set, above. We would conclude that if r is not in f, then r is in f and vice versa.
It is not possible to put P1(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.