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Cantor's diagonal argument
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{{short description|Proof in set theory}} {{Anchor|Lead}}[[Image:Diagonal argument 01 svg.svg|right|thumb|250px|An illustration of Cantor's diagonal argument (in base 2) for the existence of [[uncountable set]]s. The sequence at the bottom cannot occur anywhere in the enumeration of sequences above.]] [[File:Aplicación 2 inyectiva sobreyectiva02.svg|right|thumb|250px|An [[infinite set]] may have the same [[cardinality]] as a proper [[subset]] of itself, as the depicted [[bijection]] ''f''(''x'')=2''x'' from the [[natural numbers|natural]] to the [[even numbers]] demonstrates. Nevertheless, infinite sets of different cardinalities exist, as Cantor's diagonal argument shows.]] '''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}} 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 |surname=Gray|given=Robert|year=1994 |url=http://www.maa.org/sites/default/files/pdf/upload_library/22/Ford/Gray819-832.pdf |title=Georg Cantor and Transcendental Numbers|journal=[[American Mathematical Monthly]]|volume=101|issue=9|pages=819–832 |doi=10.2307/2975129|jstor=2975129}}</ref><ref name="Bloch2011">{{cite book|last1=Bloch|first1=Ethan D.|title=The Real Numbers and Real Analysis|url=https://archive.org/details/realnumbersreala00edbl|url-access=limited|date=2011|publisher=Springer|location=New York|isbn=978-0-387-72176-7|page=[https://archive.org/details/realnumbersreala00edbl/page/n458 429]}}</ref> 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 |publisher=Stanford encyclopedia of philosophy}}</ref><ref>{{cite book|author=Bertrand Russell|title=Principles of mathematics|pages=363–366|publisher=Norton|year=1931}}</ref> and [[Richard's paradox]].<ref name="Simmons1993"/>{{rp|27}} == Uncountable set == Cantor considered the set ''T'' of all infinite [[sequences]] of [[binary digits]] (i.e. each digit is zero or one).<ref group=note>Cantor used "''m'' and "''w''" instead of "0" and "1", "''M''" instead of "''T''", and "''E''<sub>''i''</sub>" instead of "''s''<sub>''i''</sub>".</ref> He begins with a [[constructive proof]] of the following [[Lemma (mathematics)|lemma]]: :If ''s''<sub>1</sub>, ''s''<sub>2</sub>, ... , ''s''<sub>''n''</sub>, ... is any enumeration of elements from ''T'',<ref group=note>Cantor does not assume that every element of ''T'' is in this enumeration.</ref> then an element ''s'' of ''T'' can be constructed that doesn't correspond to any ''s''<sub>''n''</sub> in the enumeration. The proof starts with an enumeration of elements from ''T'', for example :{| |- | ''s''<sub>1</sub> = || (0, || 0, || 0, || 0, || 0, || 0, || 0, || ...) |- | ''s''<sub>2</sub> = || (1, || 1, || 1, || 1, || 1, || 1, || 1, || ...) |- | ''s''<sub>3</sub> = || (0, || 1, || 0, || 1, || 0, || 1, || 0, || ...) |- | ''s''<sub>4</sub> = || (1, || 0, || 1, || 0, || 1, || 0, || 1, || ...) |- | ''s''<sub>5</sub> = || (1, || 1, || 0, || 1, || 0, || 1, || 1, || ...) |- | ''s''<sub>6</sub> = || (0, || 0, || 1, || 1, || 0, || 1, || 1, || ...) |- | ''s''<sub>7</sub> = || (1, || 0, || 0, || 0, || 1, || 0, || 0, || ...) |- | ... |} 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''-th digit as complementary to the ''n''-th digit of ''s''<sub>''n''</sub>. For the example above, this yields :{| |- | ''s''<sub>1</sub> || = || (<u>'''0'''</u>, || 0, || 0, || 0, || 0, || 0, || 0, || ...) |- | ''s''<sub>2</sub> || = || (1, || <u>'''1'''</u>, || 1, || 1, || 1, || 1, || 1, || ...) |- | ''s''<sub>3</sub> || = || (0, || 1, || <u>'''0'''</u>, || 1, || 0, || 1, || 0, || ...) |- | ''s''<sub>4</sub> || = || (1, || 0, || 1, || <u>'''0'''</u>, || 1, || 0, || 1, || ...) |- | ''s''<sub>5</sub> || = || (1, || 1, || 0, || 1, || <u>'''0'''</u>, || 1, || 1, || ...) |- | ''s''<sub>6</sub> || = || (0, || 0, || 1, || 1, || 0, || <u>'''1'''</u>, || 1, || ...) |- | ''s''<sub>7</sub> || = || (1, || 0, || 0, || 0, || 1, || 0, || <u>'''0'''</u>, || ...) |- | ... |- | |- | ''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''-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]] to show that: :The set ''T'' is uncountable. The proof starts by assuming that ''T'' is [[countable set#Definition|countable]]. Then all its elements can be written in an enumeration ''s''<sub>1</sub>, ''s''<sub>2</sub>, ... , ''s''<sub>''n''</sub>, ... . 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.<ref name="Cantor.1891"/> === Real numbers === 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''' 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 {{nowrap|''f''{{space|hair}}(''t'') {{=}} 0.''t''}}, is an injection because it maps different strings to different numbers.<ref group=note>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.</ref> 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]]-''b'' number: 0.0111...<sub>''b''</sub>. This leads to the family of functions: {{nowrap|''f''<sub>''b''</sub>{{space|hair}}(''t'') {{=}} 0.''t''<sub>''b''</sub>}}. The functions {{nowrap|''f''{{space|hair}}<sub>''b''</sub>(''t'')}} are injections, except for {{nowrap|''f''{{space|hair}}<sub>2</sub>(''t'')}}. This function will be modified to produce a bijection between ''T'' and '''R'''. {| 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}}. 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. To modify {{nowrap|''f''<sub>2</sub>{{space|hair}}(''t'')}}, observe that it is a bijection except for a countably infinite subset of (0, 1) and a countably infinite subset of ''T''. It is not a bijection for the numbers in (0, 1) that have two [[binary expansion]]s. These are called [[dyadic rational|dyadic]] numbers and have the form {{nowrap|''m''{{space|hair}}/{{space|hair}}2<sup>''n''</sup>}} where ''m'' is an odd integer and ''n'' is a natural number. Put these numbers in the sequence: ''r'' = (1/2, 1/4, 3/4, 1/8, 3/8, 5/8, 7/8, ...). Also, {{nowrap|''f''<sub>2</sub>{{space|hair}}(''t'')}} is not a bijection to (0, 1) for the strings in ''T'' appearing after the [[binary point]] in the binary expansions of 0, 1, and the numbers in sequence ''r''. Put these eventually-constant strings in the sequence: ''s'' = ({{color|#808080|000}}..., {{color|#808080|111}}..., 1{{color|#808080|000}}..., 0{{color|#808080|111}}..., 01{{color|#808080|000}}..., 00{{color|#808080|111}}..., 11{{color|#808080|000}}..., 10{{color|#808080|111}}..., ...). Define the bijection ''g''(''t'') from ''T'' to (0, 1): If ''t'' is the ''n''<sup>th</sup> string in sequence ''s'', let ''g''(''t'') be the ''n''<sup>th</sup> number in sequence ''r''{{space|hair}}; otherwise, ''g''(''t'') = 0.''t''<sub>2</sub>. To construct a bijection from ''T'' to '''R''', start with the [[trigonometric functions|tangent function]] tan(''x''), which is a bijection from (−π/2, π/2) to '''R''' (see the figure shown on the right). Next observe that the [[linear function]] ''h''(''x'') = {{nowrap|π''x'' – π/2}} is a bijection from (0, 1) to (−π/2, π/2) (see the figure shown on the left). The [[function composition|composite function]] tan(''h''(''x'')) = {{nowrap|tan(π''x'' – π/2)}} is a bijection from (0, 1) to '''R'''. Composing this function with ''g''(''t'') produces the function tan(''h''(''g''(''t''))) = {{nowrap|tan(π''g''(''t'') – π/2)}}, which is a bijection from ''T'' to '''R'''. |} ===General sets=== [[File:Diagonal argument powerset svg.svg|thumb|250px|Illustration of the generalized diagonal argument: The set <math>T = \{n \in \mathbb{N}: n \not\in f(n)\}</math> at the bottom cannot occur anywhere in the [[Range of a function|range]] of <math>f:\mathbb{N}\to\mathcal{P}(\mathbb{N})</math>. The example mapping ''f'' happens to correspond to the example enumeration ''s'' in the picture [[#Lead|above]].]] 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''). 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 (mathematics)|image]] of ''f''. As a candidate consider the set : <math>T = \{ s \in S : s \notin 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''); see picture. For a more complete account of this proof, see [[Cantor's theorem]]. ==Consequences== ===Ordering of cardinals=== With equality defined as the existence of a bijection between their underlying sets, Cantor also defines binary predicate of cardinalities <math>|S|</math> and <math>|T|</math> in terms of the [[Cardinality#Comparing_sets|existence of injections]] between <math>S</math> and <math>T</math>. It has the properties of a [[preorder]] and is here written "<math>\le</math>". One can embed the naturals into the binary sequences, thus proving various ''injection existence'' statements explicitly, so that in this sense <math>|{\mathbb N}|\le|2^{\mathbb N}|</math>, where <math>2^{\mathbb N}</math> denotes the function space <math>{\mathbb N}\to\{0,1\}</math>. 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 <math>|{\mathbb N}|<|2^{\mathbb N}|</math>, where "<math><</math>" 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 [[Von Neumann cardinal assignment|assigned]] [[ordinal numbers|ordinals]]). Also <math>|S|<|{\mathcal P}(S)|</math> in this sense, as has been shown, and at the same time it is the case that <math>\neg(|{\mathcal P}(S)|\le|S|)</math>, for all sets <math>S</math>. Assuming the [[law of excluded middle]], [[characteristic functions]] surject onto powersets, and then <math>|2^S|=|{\mathcal P}(S)|</math>. So the uncountable <math>2^{\mathbb N}</math> is also not enumerable and it can also be mapped onto <math>{\mathbb N}</math>. 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 <math>{\mathbb N}</math> is then in bijection with <math>{\mathbb N}</math> itself, and every [[subcountable]] set (a property in terms of surjections) is then already countable, i.e. in the surjective image of <math>{\mathbb N}</math>. In this context the possibilities are then exhausted, making "<math>\le</math>" a [[partial order|non-strict partial order]], or even a [[total order]] when assuming [[axiom of choice|choice]]. 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 <math>S</math> were the set of all sets, then <math>{\mathcal P}(S)</math> would at the same time be bigger than <math>S</math> and a subset of <math>S</math>. ====In the absence of excluded middle==== Also in [[Constructivism (mathematics)|constructive mathematics]], there is no surjection from the full domain <math>{\mathbb N}</math> onto the space of functions <math>{\mathbb N}^{\mathbb N}</math> or onto the collection of subsets <math>{\mathcal P}({\mathbb N})</math>, which is to say these two collections are uncountable. Again using "<math><</math>" for proven injection existence in conjunction with bijection absence, one has <math>{\mathbb N}<2^{\mathbb N}</math> and <math>S<{\mathcal P}(S)</math>. Further, <math>\neg({\mathcal P}(S)\le S)</math>, as previously noted. Likewise, <math>2^{\mathbb N}\le{\mathbb N}^{\mathbb N}</math>, <math>2^S\le{\mathcal P}(S)</math> and of course <math>S\le S</math>, 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.<ref>{{Cite arXiv|eprint=1904.09193|title=Cantor-Bernstein implies Excluded Middle|class=math.LO|last1=Pradic|first1=Cécilia|last2=Brown|first2=Chad E.|year=2019}}</ref> 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 set|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 <math>2^{\mathbb N}</math> or <math>{\mathbb N}^{\mathbb N}</math> may be asserted to be [[subcountability|subcountable]].<ref>{{citation | last = Bell | first = John L. | author-link = John Lane Bell | editor-last = Link | editor-first = Godehard | contribution = Russell's paradox and diagonalization in a constructive context | contribution-url = https://publish.uwo.ca/~jbell/russ.pdf | mr = 2104745 | pages = 221–225 | publisher = de Gruyter, Berlin | series = De Gruyter Series in Logic and its Applications | title = One hundred years of Russell's paradox | 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]", 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=== [[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 versus NP|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 [[unrestricted comprehension|naive axiom scheme of comprehension]] is modified to avoid the paradoxes by introducing a kind of "local" [[type theory]]. In this axiom scheme, :{ ''s'' ∈ ''S'': ''s'' ∉ ''f''(''s'') } 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 :{ ''s'' ∈ ''S'': ''s'' ∉ ''f''({''s''}) } ''is'' a set in NF. In which case, if '''''P'''''<sub>1</sub>(''S'') is the set of one-element subsets of ''S'' and ''f'' is a proposed bijection from '''''P'''''<sub>1</sub>(''S'') to '''''P'''''(''S''), one is able to use [[proof by contradiction]] to prove that |'''''P'''''<sub>1</sub>(''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''({''r''}) coincides with the modified diagonal set, above. We would conclude that if ''r'' is not in ''f''({''r''}), then ''r'' is in ''f''({''r''}) and vice versa. It is ''not'' possible to put '''''P'''''<sub>1</sub>(''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 group=note/> ==References== <references/> == External links == * [http://www.mathpages.com/home/kmath371.htm Cantor's Diagonal Proof] at MathPages * {{MathWorld|title=Cantor Diagonal Method|id=CantorDiagonalMethod}} {{Set theory}} {{Mathematical logic}} {{Use dmy dates|date=April 2017}} {{DEFAULTSORT:Cantor's Diagonal Argument}} [[Category:Set theory]] [[Category:Theorems in the foundations of mathematics]] [[Category:Mathematical proofs]] [[Category:Infinity]] [[Category:Arguments]] [[Category:Cardinal numbers]] [[Category:Georg Cantor]]
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