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'''Cantor's diagonal argument''' (among various similar names<ref group="note">the '''diagonalisation argument''', the '''diagonal slash argument''', the '''anti-diagonal argument''', the '''diagonal method''', and '''Cantor's diagonalization proof'''</ref>) is a [[mathematical proof]] that there are [[infinite set]]s which cannot be put into [[bijection|one-to-one correspondence]] with the infinite set of [[natural number]]s{{snd}}informally, that there are [[Set (mathematics)|set]]s which in some sense contain more elements than there are positive integers. Such sets are now called [[uncountable set]]s, and the size of infinite sets is treated by the theory of [[cardinal number]]s, which Cantor began.
[[Georg Cantor]] published this proof in 1891,<ref name="Cantor.1891">{{cite journal |author=Georg Cantor |title=Ueber eine elementare Frage der Mannigfaltigkeitslehre |journal=[[Jahresbericht der Deutschen Mathematiker-Vereinigung]] |volume=1 |pages=75–78 |year=1891 |url=https://www.digizeitschriften.de/dms/img/?PID=GDZPPN002113910&physid=phys84#navi |archive-date=3 January 2023 |access-date=11 June 2018 |archive-url=https://web.archive.org/web/20230103204747/https://www.digizeitschriften.de/dms/img/?PID=GDZPPN002113910&physid=phys84#navi |url-status=live }} English translation: {{cite book |editor-last=Ewald |editor-first=William B. |title=From Immanuel Kant to David Hilbert: A Source Book in the Foundations of Mathematics, Volume 2 |publisher=Oxford University Press |pages=920–922 |year=1996 |isbn=0-19-850536-1}}</ref><ref name="Simmons1993">{{cite book|author=Keith Simmons| author-link=Keith Simmons (philosopher)| title=Universality and the Liar: An Essay on Truth and the Diagonal Argument|url=https://books.google.com/books?id=wEj3Spept0AC&pg=PA20|date=30 July 1993|publisher=Cambridge University Press|isbn=978-0-521-43069-2}}</ref>{{rp|20–}}<ref name="Rubin1976">{{cite book|last1=Rudin|first1=Walter|title=Principles of Mathematical Analysis|date=1976|publisher=McGraw-Hill|___location=New York|isbn=0070856133|page=[https://archive.org/details/principlesofmath00rudi/page/30 30]|edition=3rd|url-access=registration|url=https://archive.org/details/principlesofmath00rudi/page/30}}</ref> but it was not [[Cantor's first uncountability proof|his first proof]] of the uncountability of the [[real number]]s, which appeared in 1874.<ref>{{Citation
However, it demonstrates a general technique that has since been used in a wide range of proofs,<ref>{{cite book |title=The Logic of Infinity |edition=illustrated |first1=Barnaby |last1=Sheppard |publisher=Cambridge University Press |year=2014 |isbn=978-1-107-05831-6 |page=73 |url=https://books.google.com/books?id=RXzsAwAAQBAJ}} [https://books.google.com/books?id=RXzsAwAAQBAJ&pg=PA73 Extract of page 73]</ref> including the first of [[Gödel's incompleteness theorems]]<ref name="Simmons1993"/> and Turing's answer to the ''[[Entscheidungsproblem]]''. Diagonalization arguments are often also the source of contradictions like [[Russell's paradox]]<ref>{{cite book|url=http://plato.stanford.edu/entries/russell-paradox|title=Russell's paradox|year=2021
== Uncountable set ==
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| ...
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Next, a sequence ''s'' is constructed by choosing the 1st digit as [[Ones' complement|complementary]] to the 1st digit of ''s''<sub>''1''</sub> (swapping '''0'''s for '''1'''s and vice versa), the 2nd digit as complementary to the 2nd digit of ''s''<sub>''2''</sub>, the 3rd digit as complementary to the 3rd digit of ''s''<sub>''3''</sub>, and generally for every ''n'', the ''n''
:{|
|-
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| ''s'' || = || (<u>'''1'''</u>, || <u>'''0'''</u>, || <u>'''1'''</u>, || <u>'''1'''</u>, || <u>'''1'''</u>, || <u>'''0'''</u>, || <u>'''1'''</u>, || ...)
|}
By construction, ''s'' is a member of ''T'' that differs from each ''s''<sub>''n''</sub>, since their ''n''
Hence, ''s'' cannot occur in the enumeration.
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The uncountability of the [[real number]]s was already established by [[Cantor's first uncountability proof]], but it also follows from the above result. To prove this, an [[injective function|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 (mathematics)|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 <math>\mathfrak{c}</math> or <math>2^{\aleph_0}</math>.
An injection from ''T'' to '''R'''
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 [[radix|base]]
{| class="wikitable collapsible collapsed"
! Construction of a bijection between ''T'' and '''R'''
|- style="text-align: left; vertical-align: top"
| {{multiple image|total_width=200|image1=Linear transformation svg.svg|width1=106|height1=159|caption1=The function ''h'': (0,1) → (−π/2, π/2)|image2=Tangent one period.svg|width2=338|height2=580|caption2=The function tan: (−π/2, π/2) → '''R'''}}
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 [[irrational number|irrational]]s 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.<ref>See page 254 of {{Citation|author=Georg Cantor|title=Ein Beitrag zur Mannigfaltigkeitslehre|url=http://www.digizeitschriften.de/dms/img/?PID=GDZPPN002156806|volume=84|pages=242–258|journal=Journal für die Reine und Angewandte Mathematik|year=1878|access-date=17 August 2017|archive-date=6 November 2018|archive-url=https://web.archive.org/web/20181106172259/http://www.digizeitschriften.de/dms/img/?PID=GDZPPN002156806|url-status=live}}. This proof is discussed in {{Citation|author=Joseph Dauben|title=Georg Cantor: His Mathematics and Philosophy of the Infinite|publisher=Harvard University Press|year=1979|isbn=0-674-34871-0}}, pp. 61–62, 65. On page 65, Dauben proves a result that is stronger than Cantor's. He lets "''φ<sub>ν</sub>'' denote any sequence of rationals in [0, 1]." Cantor lets ''φ<sub>ν</sub>'' denote a sequence [[Enumeration|enumerating]] the rationals in [0, 1], which is the kind of sequence needed for his construction of a bijection between [0, 1] and the irrationals in (0, 1).</ref>
Cantor's method can be used to modify the function {{nowrap|''f''{{space|hair}}<sub>2</sub>(''t'') {{=}} 0.''t''<sub>2</sub>}} to produce a bijection from ''T'' to (0, 1). Because some numbers have two binary expansions, {{nowrap|''f''{{space|hair}}<sub>2</sub>(''t'')}} is not even [[injective function|injective]]. For example, {{nowrap|''f''{{space|hair}}<sub>2</sub>(1000...) {{=}}}} 0.1000...<sub>2</sub> = 1/2 and {{nowrap|''f''{{space|hair}}<sub>2</sub>(0111...) {{=}}}} 0.0111...<sub>2</sub> = {{nowrap|[[Infinite series|1/4 + 1/8 + 1/16 + ...]] {{=}}}} 1/2, so both 1000... and 0111... map to the same number, 1/2.
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A generalized form of the diagonal argument was used by Cantor to prove [[Cantor's theorem]]: for every [[Set (mathematics)|set]] ''S'', the [[power set]] of ''S''—that is, the set of all [[subset]]s of ''S'' (here written as '''''P'''''(''S''))—cannot be in [[bijection]] with ''S'' itself. This proof proceeds as follows:
Let ''f'' be any [[Function (mathematics)|function]] from ''S'' to '''''P'''''(''S'').
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,
▲:<math>T = \{ s \in S : s \not\in f(s) \}</math>.
▲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]].
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| volume = 6
| year = 2004}}</ref><ref>Rathjen, M. "[http://www1.maths.leeds.ac.uk/~rathjen/acend.pdf Choice principles in constructive and classical set theories]", Proceedings of the Logic Colloquium, 2002</ref>
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 <math>2^{\mathbb N}</math> or <math>{\mathbb N}^{\mathbb N}</math> into <math>{\mathbb N}</math> is here possible as well.<ref>Bauer, A. "[http://math.andrej.com/wp-content/uploads/2011/06/injection.pdf An injection from N^N to N] {{Webarchive|url=https://web.archive.org/web/20211127195842/http://math.andrej.com/wp-content/uploads/2011/06/injection.pdf |date=27 November 2021 }}", 2011</ref> So the cardinal relation fails to be [[Antisymmetric relation|antisymmetric]]. Consequently, also in the presence of function space sets that are even classically uncountable, [[intuitionist]]s do not accept this relation to constitute a hierarchy of transfinite sizes.<ref>{{cite book |title=Mathematics and Logic in History and in Contemporary Thought |author=Ettore Carruccio |publisher=Transaction Publishers |year=2006 |page=354 |isbn=978-0-202-30850-0}}</ref>
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 [[Axiom schema of predicative separation|Predicative Separation]].
In a set theory, theories of mathematics are [[Model theory|modeled]]. Weaker logical axioms mean fewer constraints and so allow for a richer class of models. A set may be identified as a [[Construction of the real numbers|model of the field of real numbers]] when it fulfills some [[Tarski's axiomatization of the reals|axioms of real numbers]] or a [[Constructive analysis|constructive rephrasing]] thereof. Various models have been studied, such as the [[Construction_of_the_real_numbers#Construction_from_Cauchy_sequences|Cauchy reals]] or the [[Dedekind cut|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, [[Effective_topos#Realizability_topoi|variants]] of the Dedekind reals can be countable<ref>{{Cite arXiv|eprint=2404.01256|title=The Countable Reals|class=math.LO|last1=Bauer|last2=Hanson|year=2024}}</ref> 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_sequence#Completeness|Cauchy-complete]]<ref>Robert S. Lubarsky, [https://arxiv.org/pdf/1510.00639.pdf ''On the Cauchy Completeness of the Constructive Cauchy Reals''], July 2015</ref> 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===
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==See also==
*[[Cantor's first uncountability proof]]
*[[Continuum hypothesis]]
*[[Controversy over Cantor's theory]]
*[[Diagonal lemma]]
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