Cantor's diagonal argument

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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 conce ...
, 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 of ...
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 proo ...
that there are
infinite set In set theory, an infinite set is a set that is not a finite set. Infinite sets may be countable or uncountable. Properties The set of natural numbers (whose existence is postulated by the axiom of infinity) is infinite. It is the only s ...
s which cannot be put into
one-to-one correspondence 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 ...
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 nu ...
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. T ...
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. Every ...
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 state ...
''. Diagonalization arguments are often also the source of contradictions like
Russell's paradox In mathematical logic, Russell's paradox (also known as Russell's antinomy) is a set-theoretic paradox discovered by the British philosopher and mathematician Bertrand Russell in 1901. Russell's paradox shows that every set theory that contains ...
and
Richard's paradox In logic, Richard's paradox is a semantical antinomy of set theory and natural language first described by the French mathematician Jules Richard in 1905. The paradox is ordinarily used to motivate the importance of distinguishing carefully betwee ...
.

# 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 called t ...
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 In mathematics, a constructive proof is a method of proof that demonstrates the existence of a mathematical object by creating or providing a method for creating the object. This is in contrast to a non-constructive proof (also known as an existenc ...
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 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 known ...
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 numbers ...
. 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. Every ...
s was already established by
Cantor's first uncountability proof Cantor's first set theory article contains Georg Cantor's first theorems of transfinite set theory, which studies infinite sets and their properties. One of these theorems is his "revolutionary discovery" that the set of all real numbers is unco ...
, but it also follows from the above result. To prove this, an
injection Injection or injected may refer to: Science and technology * Injective function, a mathematical function mapping distinct arguments to distinct values * Injection (medicine), insertion of liquid into the body with a syringe * Injection, in broadca ...
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 $\mathfrak$ 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 numeral ...
, 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 be ...
: for every
set Set, The Set, SET or SETS may refer to: Science, technology, and mathematics Mathematics *Set (mathematics), a collection of elements *Category of sets, the category whose objects and morphisms are sets and total functions, respectively Electro ...
''S'', the
power set In mathematics, the power set (or powerset) of a set is the set of all subsets of , including the empty set and itself. In axiomatic set theory (as developed, for example, in the ZFC axioms), the existence of the power set of any set is post ...
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-oriente ...
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 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 be ...
.

# Consequences

## Ordering of cardinals

Cantor defines an
order relation Order theory is a branch of mathematics that investigates the intuitive notion of order using binary relations. It provides a formal framework for describing statements such as "this is less than that" or "this precedes that". This article int ...
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 In set theory, a universal set is a set which contains all objects, including itself. In set theory as usually formulated, it can be proven in multiple ways that a universal set does not exist. However, some non-standard variants of set theory inc ...
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 noncontradi ...
, every
subcountable In constructive mathematics, a collection X is subcountable if there exists a partial surjection from the natural numbers onto it. This may be expressed as \exists (I\subseteq).\, \exists f.\, (f\colon I\twoheadrightarrow X), where f\colon I\twohe ...
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 th ...
, 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 f ...
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 synta ...
, i.e. the right set of counting numbers for the subcountable sets may not be
recursive Recursion (adjective: ''recursive'') occurs when a thing is defined in terms of itself or of its type. Recursion is used in a variety of disciplines ranging from linguistics to logic. The most common application of recursion is in mathematics ...
. 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 In constructive mathematics, a collection X is subcountable if there exists a partial surjection from the natural numbers onto it. This may be expressed as \exists (I\subseteq).\, \exists f.\, (f\colon I\twoheadrightarrow X), where f\colon I\twohe ...
, 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 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 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 ...
.

## Diagonalization in broader context

Russell's paradox In mathematical logic, Russell's paradox (also known as Russell's antinomy) is a set-theoretic paradox discovered by the British philosopher and mathematician Bertrand Russell in 1901. Russell's paradox shows that every set theory that contains ...
has shown that naive set theory, based on an
unrestricted comprehension In many popular versions of axiomatic set theory, the axiom schema of specification, also known as the axiom schema of separation, subset axiom scheme or axiom schema of restricted comprehension is an axiom schema. Essentially, it says that any ...
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 In computational complexity theory, a complexity class is a set of computational problems of related resource-based complexity. The two most commonly analyzed resources are time and memory. In general, a complexity class is defined in terms of a ...
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 Willard Van Orman Quine (; known to his friends as "Van"; June 25, 1908 – December 25, 2000) was an American philosopher and logician in the analytic tradition, recognized as "one of the most influential philosophers of the twentieth century". ...
'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 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 fou ...
. 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 known ...
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 Cantor's first set theory article contains Georg Cantor's first theorems of transfinite set theory, which studies infinite sets and their properties. One of these theorems is his "revolutionary discovery" that the set of all real numbers is unco ...
* Controversy over Cantor's theory *
Diagonal lemma In mathematical logic, the diagonal lemma (also known as diagonalization lemma, self-reference lemma or fixed point theorem) establishes the existence of self-referential sentences in certain formal theories of the natural numbers—specificall ...