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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.]]
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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| ...
|}
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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===General sets===
[[File:Diagonal argument powerset svg.svg|thumb|250px|Illustration of the generalized diagonal argument: The set
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'').
: <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,
▲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===
With equality defined as the existence of a bijection between their underlying sets, Cantor also defines
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>.
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.
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=
| last = Bell | first = John L. | author-link = John Lane Bell
| editor-last = Link | editor-first = Godehard
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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
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
===Diagonalization in broader context===
[[Russell's paradox]] has shown that
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]].
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==See also==
*[[Cantor's first uncountability proof]]
*[[Continuum hypothesis]]
*[[Controversy over Cantor's theory]]
*[[Diagonal lemma]]
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