View source for Module:Cantor's diagonal argument

{{short description|Proof in set theory}}
{{about-distinguish-text|a concept in set and number theory|matrix diagonalization. See [[diagonalization (disambiguation)]] for several other uses of the term in mathematics}}

{{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.]]

In [[set theory]], '''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]] as 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.<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> Such sets are now known as [[uncountable set]]s, and the size of infinite sets is now treated by the theory of [[cardinal number]]s which Cantor began.

The diagonal argument was not Cantor's [[Cantor's first uncountability proof|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''<sup>th</sup> digit as complementary to the ''n''<sup>th</sup> 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''<sup>th</sup> 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''&nbsp;=&nbsp;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,&nbsp;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&ndash;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&ndash;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,&nbsp;1]." Cantor lets ''φ<sub>ν</sub>'' denote a sequence [[Enumeration|enumerating]] the rationals in [0,&nbsp;1], which is the kind of sequence needed for his construction of a bijection between [0,&nbsp;1] and the irrationals in (0,&nbsp;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,&nbsp;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>&nbsp;= 1/2 and {{nowrap|''f''{{space|hair}}<sub>2</sub>(0111...) {{=}}}} 0.0111...<sub>2</sub>&nbsp;= {{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,&nbsp;1) and a countably infinite subset of ''T''. It is not a bijection for the numbers in (0,&nbsp;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''&nbsp;= (1/2, 1/4, 3/4, 1/8, 3/8, 5/8, 7/8,&nbsp;...). Also, {{nowrap|''f''<sub>2</sub>{{space|hair}}(''t'')}} is not a bijection to (0,&nbsp;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''&nbsp;= ({{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}}...,&nbsp;...). Define the bijection ''g''(''t'') from ''T'' to (0,&nbsp;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'')&nbsp;=&nbsp;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,&nbsp;π/2) to '''R''' (see the figure shown on the right). Next observe that the [[linear function]] ''h''(''x'')&nbsp;= {{nowrap|π''x'' – π/2}} is a bijection from (0,&nbsp;1) to (−π/2,&nbsp;π/2) (see the figure shown on the left). The [[function composition|composite function]] tan(''h''(''x''))&nbsp;= {{nowrap|tan(π''x'' – π/2)}} is a bijection from (0,&nbsp;1) to '''R'''. Composing this function with ''g''(''t'') produces the function tan(''h''(''g''(''t'')))&nbsp;= {{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 ''T'' = {''n''∈<math>\mathbb{N}</math>: ''n''∉''f''(''n'')} at the bottom cannot occur anywhere in the range of ''f'':[[natural number|<math>\mathbb{N}</math>]]→[[power set|'''''P''''']](<math>\mathbb{N}</math>). The example mapping ''f'' happens to correspond to the example enumeration ''s'' in the [[#Lead|above]] picture.]]
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 ''f'' cannot be [[surjective]]. That 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:

:''T'' = { ''s'' ∈ ''S'': ''s'' ∉ ''f''(''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,  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]].

==Consequences==
===Ordering of cardinals===
Cantor defines an [[Total order|order relation]] of cardinalities, e.g. <math>|S|</math> and <math>|T|</math>, in terms of the existence of injections between the underlying sets, <math>S</math> and <math>T</math>. The argument in the previous paragraph then proved that sets such as <math>{\mathbb N}\to\{0,1\}</math> are uncountable, which is to say <math>\neg(|{\mathbb N}\to\{0,1\}|\le|{\mathbb N}|)</math>, and we can embed the naturals into the function space, so that we have that <math>|{\mathbb N}|<|{\mathbb N}\to\{0,1\}|</math>. 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]], every [[subcountable]] set (a property in terms of surjections) is also already countable. 
 
In [[Constructivism (mathematics)|Constructive mathematics]], it is harder or impossible to order ordinals and also cardinals. 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=Pierre|last2=Brown|first2=Chad E.|year=2019}}</ref> Therefore, [[intuitionist]]s do not accept the theorem about the cardinal ordering.<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> 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]], i.e. the right set of counting numbers for the subcountable sets may not be [[Recursive set|recursive]]. 
In a set theory, theories of mathematics are [[Model theory|modeled]]. For example, in set theory, "the" set of real numbers [[Construction of the real numbers|is identified]] as a set that fulfills some [[Tarski's axiomatization of the reals|axioms of real numbers]]. Various models have been studied, such as the [[Dedekind cut|Dedekind reals]] or the [[Cauchy sequence|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 <math>{\mathbb N}\to\{0,1\}</math> may be asserted to be [[subcountability|subcountable]], a notion of size orthogonal to the statement <math>|{\mathbb N}|<|{\mathbb N}\to\{0,1\}|</math>.<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> In such a context, the subcountability of all sets is possible, or injections from the uncountable <math>{\mathbb N}^{\mathbb N}</math> into <math>{\mathbb N}</math>.<ref>Bauer, A. "[http://math.andrej.com/wp-content/uploads/2011/06/injection.pdf An injection from N^N to N]", 2011</ref>

===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]]. 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]].

===Diagonalization in broader context===
[[Russell's paradox]] has shown that [[naive set theory]], 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]] 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'')| &lt; |'''''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]]
*[[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}}
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{{Use dmy dates|date=April 2017}}

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