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In
set theory Set theory is the branch of mathematical logic that studies sets, which can be informally described as collections of objects. Although objects of any kind can be collected into a set, set theory, as a branch of mathematics, is mostly concer ...
, Cantor's diagonal argument, also called the diagonalisation argument, the diagonal slash argument, the anti-diagonal argument, the diagonal method, and Cantor's diagonalization proof, was published in 1891 by
Georg Cantor Georg Ferdinand Ludwig Philipp Cantor ( , ;  – January 6, 1918) was a German mathematician. He played a pivotal role in the creation of set theory, which has become a fundamental theory in mathematics. Cantor established the importance o ...
as a
mathematical proof A mathematical proof is an inferential argument for a mathematical statement, showing that the stated assumptions logically guarantee the conclusion. The argument may use other previously established statements, such as theorems; but every pr ...
that there are infinite sets which cannot be put into one-to-one correspondence with the infinite set of
natural number In mathematics, the natural numbers are those numbers used for counting (as in "there are ''six'' coins on the table") and ordering (as in "this is the ''third'' largest city in the country"). Numbers used for counting are called ''cardinal ...
s. English translation: Such sets are now known as
uncountable set In mathematics, an uncountable set (or uncountably infinite set) is an infinite set that contains too many elements to be countable. The uncountability of a set is closely related to its cardinal number: a set is uncountable if its cardinal num ...
s, and the size of infinite sets is now treated by the theory of
cardinal number In mathematics, cardinal numbers, or cardinals for short, are a generalization of the natural numbers used to measure the cardinality (size) of sets. The cardinality of a finite set is a natural number: the number of elements in the set. The ...
s which Cantor began. The diagonal argument was not Cantor's first proof of the uncountability of the
real number In mathematics, a real number is a number that can be used to measure a ''continuous'' one- dimensional quantity such as a distance, duration or temperature. Here, ''continuous'' means that values can have arbitrarily small variations. Ever ...
s, which appeared in 1874. However, it demonstrates a general technique that has since been used in a wide range of proofs, including the first of Gödel's incompleteness theorems and Turing's answer to the ''
Entscheidungsproblem In mathematics and computer science, the ' (, ) is a challenge posed by David Hilbert and Wilhelm Ackermann in 1928. The problem asks for an algorithm that considers, as input, a statement and answers "Yes" or "No" according to whether the stat ...

# Uncountable set

Cantor considered the set ''T'' of all infinite
sequences In mathematics, a sequence is an enumerated collection of objects in which repetitions are allowed and order matters. Like a set, it contains members (also called ''elements'', or ''terms''). The number of elements (possibly infinite) is calle ...
of binary digits (i.e. each digit is zero or one).Cantor used "''m'' and "''w''" instead of "0" and "1", "''M''" instead of "''T''", and "''E''''i''" instead of "''s''''i''". He begins with a constructive proof of the following lemma: :If ''s''1, ''s''2, ... , ''s''''n'', ... is any enumeration of elements from ''T'',Cantor does not assume that every element of ''T'' is in this enumeration. then an element ''s'' of ''T'' can be constructed that doesn't correspond to any ''s''''n'' in the enumeration. The proof starts with an enumeration of elements from ''T'', for example : Next, a sequence ''s'' is constructed by choosing the 1st digit as
complementary A complement is something that completes something else. Complement may refer specifically to: The arts * Complement (music), an interval that, when added to another, spans an octave ** Aggregate complementation, the separation of pitch-class ...
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 : By construction, ''s'' is a member of ''T'' that 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 lemma, Cantor then uses a
proof by contradiction In logic and mathematics, proof by contradiction is a form of proof that establishes the truth or the validity of a proposition, by showing that assuming the proposition to be false leads to a contradiction. Proof by contradiction is also know ...
to show that: :The set ''T'' is uncountable. The proof starts by assuming that ''T'' is
countable In mathematics, a set is countable if either it is finite or it can be made in one to one correspondence with the set of natural numbers. Equivalently, a set is ''countable'' if there exists an injective function from it into the natural numbe ...
. Then all its elements can be written in an enumeration ''s''1, ''s''2, ... , ''s''''n'', ... . 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.

## Real numbers

The uncountability of the
real number In mathematics, a real number is a number that can be used to measure a ''continuous'' one- dimensional quantity such as a distance, duration or temperature. Here, ''continuous'' means that values can have arbitrarily small variations. Ever ...
s 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 An image is a visual representation of something. It can be two-dimensional, three-dimensional, or somehow otherwise feed into the visual system to convey information. An image can be an artifact, such as a photograph or other two-dimension ...
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 In mathematics, a bijection, also known as a bijective function, one-to-one correspondence, or invertible function, is a function between the elements of two sets, where each element of one set is paired with exactly one element of the other s ...
will be constructed between ''T'' and R. Therefore, ''T'' and R have the same cardinality, which is called the "
cardinality of the continuum In set theory, the cardinality of the continuum is the cardinality or "size" of the set of real numbers \mathbb R, sometimes called the continuum. It is an infinite cardinal number and is denoted by \mathfrak c (lowercase fraktur "c") or , \mat ...
" and is usually denoted by $\mathfrak$ or $2^$. An injection from ''T'' to R is given by mapping binary strings in ''T'' to
decimal fractions The decimal numeral system (also called the base-ten positional numeral system and denary or decanary) is the standard system for denoting integer and non-integer numbers. It is the extension to non-integer numbers of the Hindu–Arabic numer ...
, 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.While 0.0111... and 0.1000... would be equal if interpreted as binary fractions (destroying injectivity), they are different when interpreted as decimal fractions, as is done by ''f''. On the other hand, since ''t'' is a binary string, the equality 0.0999... = 0.1000... of decimal fractions is not relevant here. 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.

## General sets

A generalized form of the diagonal argument was used by Cantor to prove
Cantor's theorem In mathematical set theory, Cantor's theorem is a fundamental result which states that, for any set A, the set of all subsets of A, the power set of A, has a strictly greater cardinality than A itself. For finite sets, Cantor's theorem can ...
: 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 Function or functionality may refer to: Computing * Function key, a type of key on computer keyboards * Function model, a structured representation of processes in a system * Function object or functor or functionoid, a concept of object-orien ...
from ''S'' to ''P''(''S''). It suffices to prove ''f'' cannot be
surjective In mathematics, a surjective function (also known as surjection, or onto function) is a function that every element can be mapped from element so that . In other words, every element of the function's codomain is the image of one element of ...
. That means that some member ''T'' of ''P''(''S''), i.e. some subset of ''S'', is not in the
image An image is a visual representation of something. It can be two-dimensional, three-dimensional, or somehow otherwise feed into the visual system to convey information. An image can be an artifact, such as a photograph or other two-dimension ...
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 In mathematical set theory, Cantor's theorem is a fundamental result which states that, for any set A, the set of all subsets of A, the power set of A, has a strictly greater cardinality than A itself. For finite sets, Cantor's theorem can ...
.

# Consequences

## Ordering of cardinals

Cantor defines an order relation of cardinalities, e.g. $, S,$ and $, T,$, in terms of the existence of injections between the underlying sets, $S$ and $T$. The argument in the previous paragraph then proved that sets such as $\to\$ are uncountable, which is to say $\neg\left(, \to\, \le, , \right)$, and we can embed the naturals into the function space, so that we have that $, , <, \to\,$. 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. Cantor's 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 In logic, the law of excluded middle (or the principle of excluded middle) states that for every proposition, either this proposition or its negation is true. It is one of the so-called three laws of thought, along with the law of noncontrad ...
, every subcountable set (a property in terms of surjections) is also already countable. In
Constructive mathematics In the philosophy of mathematics, constructivism asserts that it is necessary to find (or "construct") a specific example of a mathematical object in order to prove that an example exists. Contrastingly, in classical mathematics, one can prove t ...
, it is harder or impossible to order ordinals and also cardinals. For example, the
Schröder–Bernstein theorem In set theory, the Schröder–Bernstein theorem states that, if there exist injective functions and between the sets and , then there exists a bijective function . In terms of the cardinality of the two sets, this classically implies that if ...
requires the law of excluded middle. Therefore,
intuitionist In the philosophy of mathematics, intuitionism, or neointuitionism (opposed to preintuitionism), is an approach where mathematics is considered to be purely the result of the constructive mental activity of humans rather than the discovery of fu ...
s do not accept the theorem about the cardinal ordering. 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 In computability theory, Rice's theorem states that all non-trivial semantic properties of programs are undecidable. A semantic property is one about the program's behavior (for instance, does the program terminate for all inputs), unlike a synt ...
, 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 $\to\$ may be asserted to be subcountable, a notion of size orthogonal to the statement $, , <, \to\,$. In such a context, the subcountability of all sets is possible, or injections from the uncountable $^$ into .Bauer, A.
An injection from N^N to N
, 2011

## Open questions

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 In mathematics, the cardinality of a set is a measure of the number of elements of the set. For example, the set A = \ contains 3 elements, and therefore A has a cardinality of 3. Beginning in the late 19th century, this concept was generalized ...
is "between" that of the integers and that of the reals. This question leads to the famous
continuum hypothesis In mathematics, the continuum hypothesis (abbreviated CH) is a hypothesis about the possible sizes of infinite sets. It states that or equivalently, that In Zermelo–Fraenkel set theory with the axiom of choice (ZFC), this is equivalent to ...
. 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.

naive set theory Naive set theory is any of several theories of sets used in the discussion of the foundations of mathematics. Unlike axiomatic set theories, which are defined using formal logic, naive set theory is defined informally, in natural language. It d ...
, 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 In computability theory, the halting problem is the problem of determining, from a description of an arbitrary computer program and an input, whether the program will finish running, or continue to run forever. Alan Turing proved in 1936 that a ...
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.

# Version for Quine's New Foundations

The above proof fails for W. V. Quine's "
New Foundations In mathematical logic, New Foundations (NF) is an axiomatic set theory, conceived by Willard Van Orman Quine as a simplification of the theory of types of ''Principia Mathematica''. Quine first proposed NF in a 1937 article titled "New Foundation ...
" 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 mathematics, logic, and computer science, a type theory is the formal presentation of a specific type system, and in general type theory is the academic study of type systems. Some type theories serve as alternatives to set theory as a founda ...
. 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 In logic and mathematics, proof by contradiction is a form of proof that establishes the truth or the validity of a proposition, by showing that assuming the proposition to be false leads to a contradiction. Proof by contradiction is also know ...
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''() 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 ''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.

* Cantor's first uncountability proof *
Controversy over Cantor's theory In mathematical logic, the theory of infinite sets was first developed by Georg Cantor. Although this work has become a thoroughly standard fixture of classical set theory, it has been criticized in several areas by mathematicians and philosoph ...
* Diagonal lemma