Cantor's diagonal argument
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Cantor's diagonal argument (among various similar namesthe diagonalisation argument, the diagonal slash argument, the anti-diagonal argument, the diagonal method, and Cantor's diagonalization proof) is a
mathematical proof A mathematical proof is a deductive reasoning, deductive Argument-deduction-proof distinctions, argument for a Proposition, mathematical statement, showing that the stated assumptions logically guarantee the conclusion. The argument may use othe ...
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 the numbers 0, 1, 2, 3, and so on, possibly excluding 0. Some start counting with 0, defining the natural numbers as the non-negative integers , while others start with 1, defining them as the positive in ...
sinformally, that there are sets which in some sense contain more elements than there are positive integers. Such sets are now called
uncountable set In mathematics, an uncountable set, informally, 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 number is larger t ...
s, and the size of infinite sets is treated by the theory of
cardinal number In mathematics, a cardinal number, or cardinal for short, is what is commonly called the number of elements of a set. In the case of a finite set, its cardinal number, or cardinality is therefore a natural number. For dealing with the cas ...
s, which Cantor began.
Georg Cantor Georg Ferdinand Ludwig Philipp Cantor ( ; ;  – 6 January 1918) was a mathematician who played a pivotal role in the creation of set theory, which has become a foundations of mathematics, fundamental theory in mathematics. Cantor establi ...
published this proof in 1891, English translation: but it was not his 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 duration or temperature. Here, ''continuous'' means that pairs of values can have arbitrarily small differences. Every re ...
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 Gödel's incompleteness theorems are two theorems of mathematical logic that are concerned with the limits of in formal axiomatic theories. These results, published by Kurt Gödel in 1931, are important both in mathematical logic and in the phi ...
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. It asks for an algorithm that considers an inputted statement and answers "yes" or "no" according to whether it is universally valid ...
''. Diagonalization arguments are often also the source of contradictions like Russell's paradox and Richard's paradox.


Uncountable set

Cantor considered the set ''T'' of all infinite sequences 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 mathematical proof, 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 ...
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, 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. Although it is quite freely used in mathematical pr ...
to show that: :The set ''T'' is uncountable. The proof starts by assuming that ''T'' is
countable In mathematics, a Set (mathematics), set is countable if either it is finite set, 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 fro ...
. 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 duration or temperature. Here, ''continuous'' means that pairs of values can have arbitrarily small differences. Every re ...
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 or picture is a visual representation. An image can be Two-dimensional space, two-dimensional, such as a drawing, painting, or photograph, or Three-dimensional space, three-dimensional, such as a carving or sculpture. Images may be di ...
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, bijective function, or one-to-one correspondence is a function between two sets such that each element of the second set (the codomain) is the image of exactly one element of the first set (the domain). Equival ...
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, 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 (mathematics), set A, the set of all subsets of A, known as the power set of A, has a strictly greater cardinality than A itself. For finite s ...
: for every set ''S'', the power set of ''S''—that is, the set of all
subset In mathematics, a Set (mathematics), set ''A'' is a subset of a set ''B'' if all Element (mathematics), elements of ''A'' are also elements of ''B''; ''B'' is then a superset of ''A''. It is possible for ''A'' and ''B'' to be equal; if they a ...
s of ''S'' (here written as ''P''(''S''))—cannot be in
bijection In mathematics, a bijection, bijective function, or one-to-one correspondence is a function between two sets such that each element of the second set (the codomain) is the image of exactly one element of the first set (the domain). Equival ...
with ''S'' itself. This proof proceeds as follows: Let ''f'' be any function from ''S'' to ''P''(''S''). It suffices to prove that ''f'' cannot be surjective. This means that some member ''T'' of ''P''(''S''), i.e. some subset of ''S'', is not in the
image An image or picture is a visual representation. An image can be Two-dimensional space, two-dimensional, such as a drawing, painting, or photograph, or Three-dimensional space, three-dimensional, such as a carving or sculpture. Images may be di ...
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''); see 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 (mathematics), set A, the set of all subsets of A, known as the power set of A, has a strictly greater cardinality than A itself. For finite s ...
.


Consequences


Ordering of cardinals

With equality defined as the existence of a bijection between their underlying sets, Cantor also defines binary predicate of cardinalities , S, and , T, in terms of the existence of injections between S and T. It has the properties of a preorder and is here written "\le". One can embed the naturals into the binary sequences, thus proving various ''injection existence'' statements explicitly, so that in this sense , , \le, 2^, , where 2^ denotes the function space \to\. But following from the argument in the previous sections, there is ''no surjection'' and so also no bijection, i.e. the set is uncountable. For this one may write , , <, 2^, , where "<" is understood to mean the existence of an injection together with the proven absence of a bijection (as opposed to alternatives such as the negation of Cantor's preorder, or a definition in terms of assigned ordinals). Also , S, <, (S), in this sense, as has been shown, and at the same time it is the case that \neg(, (S), \le, S, ), for all sets S. Assuming the law of excluded middle, characteristic functions surject onto powersets, and then , 2^S, =, (S), . So the uncountable 2^ is also not enumerable and it can also be mapped onto . Classically, the Schröder–Bernstein theorem is valid and says that any two sets which are in the injective image of one another are in bijection as well. Here, every unbounded subset of is then in bijection with itself, and every subcountable set (a property in terms of surjections) is then already countable, i.e. in the surjective image of . In this context the possibilities are then exhausted, making "\le" a non-strict partial order, or even a total order when assuming
choice A choice is the range of different things from which a being can choose. The arrival at a choice may incorporate Motivation, motivators and Choice modelling, models. Freedom of choice is generally cherished, whereas a severely limited or arti ...
. The diagonal argument thus establishes that, 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 (S) would at the same time be bigger than S and a subset of S.


In the absence of excluded middle

Also in constructive mathematics, there is no surjection from the full domain onto the space of functions ^ or onto the collection of subsets (), which is to say these two collections are uncountable. Again using "<" for proven injection existence in conjunction with bijection absence, one has <2^ and S<(S). Further, \neg((S)\le S), as previously noted. Likewise, 2^\le^, 2^S\le(S) and of course S\le S, also in constructive set theory. 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. 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 2^ or ^ may be asserted to be subcountable. This is a notion of size that is redundant in the classical context, but otherwise need not imply countability. The existence of injections from the uncountable 2^ or ^ into is here possible as well. So the cardinal relation fails to be antisymmetric. Consequently, also in the presence of function space sets that are even classically uncountable, intuitionists do not accept this relation to constitute a hierarchy of transfinite sizes. When the axiom of powerset is not adopted, in a constructive framework even the subcountability of all sets is then consistent. That all said, in common set theories, the non-existence of a set of all sets also already follows from Predicative Separation. 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 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-completeRobert S. Lubarsky
''On the Cauchy Completeness of the Constructive Cauchy Reals''
July 2015
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.


Diagonalization in broader context

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 In computability theory (computer science), 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 for ...
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" 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 and theoretical computer science, a type theory is the formal presentation of a specific type system. Type theory is the academic study of type systems. Some type theories serve as alternatives to set theory as a foundation of ...
. 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, 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. Although it is quite freely used in mathematical pr ...
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.


See also

* Cantor's first uncountability proof * Continuum hypothesis * Controversy over Cantor's theory * Diagonal lemma


Notes


References


External links


Cantor's Diagonal Proof
at MathPages * {{DEFAULTSORT:Cantor's Diagonal Argument Set theory Theorems in the foundations of mathematics Mathematical proofs Infinity Arguments Cardinal numbers Georg Cantor