Editing Georg Cantor (section)

===Set theory===
[[File:Diagonal argument 2.svg|right|thumb|250px|An illustration of [[Cantor's diagonal argument]] for the existence of [[uncountable set]]s.<ref>This follows closely the first part of Cantor's 1891 paper.</ref> The sequence at the bottom cannot occur anywhere in the infinite list of sequences above.]]
The beginning of set theory as a branch of mathematics is often marked by the publication of [[Georg Cantor's first set theory article|Cantor's 1874 paper]],<ref name="Johnson p. 55"/> "Ueber<!--[sic; see 'Talk:Gottlob Frege#Ueber']--> eine Eigenschaft des Inbegriffes aller reellen algebraischen Zahlen" ("On a Property of the Collection of All Real Algebraic Numbers").<ref>{{Harvnb|Cantor|1874}}. English translation: [[#Ewald|Ewald 1996]], pp. 840–843.</ref> This paper was the first to provide a rigorous proof that there was more than one kind of infinity. Previously, all infinite collections had been implicitly assumed to be [[equinumerosity|equinumerous]] (that is, of "the same size" or having the same number of elements).<ref>For example, geometric problems posed by [[Galileo Galilei|Galileo]] and [[John Duns Scotus]] suggested that all infinite sets were equinumerous&nbsp;– see {{Cite journal |last=Moore |first= A. W. |date=April 1995|title=A brief history of infinity|journal=Scientific American|volume=272|issue=4|pages=112–116 (114)|doi=10.1038/scientificamerican0495-112|bibcode=1995SciAm.272d.112M}}</ref> Cantor proved that the collection of real numbers and the collection of positive [[integers]] are not equinumerous. In other words, the real numbers are not [[countable]]. His proof differs from the [[Cantor's diagonal argument|diagonal argument]] that he gave in 1891.<ref>For this, and more information on the mathematical importance of Cantor's work on set theory, see e.g., [[#Suppes|Suppes 1972]].</ref> Cantor's article also contains a new method of constructing [[transcendental number]]s. Transcendental numbers were first constructed by [[Joseph Liouville]] in 1844.<ref>Liouville, Joseph (13 May 1844). [http://bibnum.education.fr/mathematiques/theorie-des-nombres/propos-de-lexistence-des-nombres-transcendants A propos de l'existence des nombres transcendants].</ref>

Cantor established these results using two constructions. His first construction shows how to write the real [[algebraic number]]s<ref>The real algebraic numbers are the real [[root]]s of [[polynomial]] equations with [[integer]] [[coefficients]].</ref> as a [[sequence]] ''a''<sub>1</sub>, ''a''<sub>2</sub>, ''a''<sub>3</sub>,&nbsp;.... In other words, the real algebraic numbers are countable. Cantor starts his second construction with any sequence of real numbers. Using this sequence, he constructs [[nested intervals]] whose [[intersection (set theory)|intersection]] contains a real number not in the sequence. Since every sequence of real numbers can be used to construct a real not in the sequence, the real numbers cannot be written as a sequence&nbsp;– that is, the real numbers are not countable.  By applying his construction to the sequence of real algebraic numbers, Cantor produces a transcendental number. Cantor points out that his constructions prove more&nbsp;– namely, they provide a new proof of Liouville's theorem: Every interval contains infinitely many transcendental numbers.<ref>For more details on Cantor's article, see [[Georg Cantor's first set theory article]] and {{Cite journal|last=Gray|first=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|access-date=6 December 2013|archive-date=21 January 2022|archive-url=https://web.archive.org/web/20220121155859/https://www.maa.org/sites/default/files/pdf/upload_library/22/Ford/Gray819-832.pdf|url-status=dead}}. Gray (pp. 821–822) describes a computer program that uses Cantor's constructions to generate a transcendental number.</ref> Cantor's next article contains a construction that proves the set of transcendental numbers has the same "power" (see below) as the set of real numbers.<ref>Cantor's construction starts with the set of transcendentals ''T'' and removes a countable [[subset]] {''t<sub>n</sub>''} (for example, ''t<sub>n</sub>''&nbsp;=&nbsp;''[[e (mathematical constant)|e]]&nbsp;/&nbsp;n''). Call this set ''T''<sub>0</sub>. Then ''T''&nbsp;= ''T''<sub>0</sub>&nbsp;∪&nbsp;{''t<sub>n</sub>''}&nbsp;= ''T''<sub>0</sub>&nbsp;∪&nbsp;{''t''<sub>2''n''-1</sub>}&nbsp;∪&nbsp;{''t''<sub>2''n''</sub>}. The set of reals '''R'''&nbsp;= ''T''&nbsp;∪&nbsp;{''a<sub>n</sub>''}&nbsp;= ''T''<sub>0</sub>&nbsp;∪&nbsp;{''t<sub>n</sub>''}&nbsp;∪&nbsp;{''a<sub>n</sub>''} where ''a<sub>n</sub>'' is the sequence of real algebraic numbers.  So both ''T'' and '''R''' are the union of three [[pairwise disjoint]] sets: ''T''<sub>0</sub> and two countable sets. A one-to-one correspondence between ''T'' and '''R''' is given by the function: ''f''(''t'')&nbsp;=&nbsp;''t'' if ''t''&nbsp;∈&nbsp;''T''<sub>0</sub>, ''f''(''t''<sub>2''n''-1</sub>)&nbsp;=&nbsp;''t<sub>n</sub>'', and ''f''(''t''<sub>2''n''</sub>)&nbsp;=&nbsp;''a<sub>n</sub>''. Cantor actually applies his construction to the irrationals rather than the transcendentals, but he knew that it applies to any set formed by removing countably many numbers from the set of reals ({{harvnb|Cantor|1879|p=4}}).</ref>

Between 1879 and 1884, Cantor published a series of six articles in ''[[Mathematische Annalen]]'' that together formed an introduction to his set theory. At the same time, there was growing opposition to Cantor's ideas, led by Leopold Kronecker, who admitted mathematical concepts only if they could be constructed in a [[finitism|finite]] number of steps from the natural numbers, which he took as intuitively given. For Kronecker, Cantor's hierarchy of infinities was inadmissible, since accepting the concept of [[actual infinity]] would open the door to paradoxes which would challenge the validity of mathematics as a whole.<ref name="popeleo">[[#Dauben1977|Dauben 1977]], p. 89.</ref> Cantor also introduced the [[Cantor set]] during this period.

The fifth paper in this series, "'''''Grundlagen einer allgemeinen Mannigfaltigkeitslehre"''''' ("''Foundations of a General Theory of Aggregates"''), published in 1883,<ref>{{harvnb|Cantor|1883}}.</ref> was the most important of the six and was also published as a separate [[monograph]]. It contained Cantor's reply to his critics and showed how the [[transfinite number]]s were a systematic extension of the natural numbers. It begins by defining [[well-order]]ed sets. [[Ordinal number]]s are then introduced as the order types of well-ordered sets. Cantor then defines the addition and multiplication of the [[cardinal number|cardinal]] and ordinal numbers. In 1885, Cantor extended his theory of order types so that the ordinal numbers simply became a special case of order types.

In 1891, he published a paper containing his elegant "diagonal argument" for the existence of an uncountable set. He applied the same idea to prove [[Cantor's theorem]]: the [[cardinality]] of the power set of a set ''A'' is strictly larger than the cardinality of ''A''. This established the richness of the hierarchy of infinite sets, and of the [[cardinal arithmetic|cardinal]] and [[ordinal arithmetic]] that Cantor had defined. His argument is fundamental in the solution of the [[Halting problem]] and the proof of [[Gödel's first incompleteness theorem]]. Cantor wrote on the [[Goldbach conjecture]] in 1894.

[[File:Passage with the set definition of Georg Cantor.png|thumb|Passage of Georg Cantor's article with his set definition]]
In 1895 and 1897, Cantor published a two-part paper in ''[[Mathematische Annalen]]'' under [[Felix Klein]]'s editorship; these were his last significant papers on set theory.<ref>{{harvtxt|Cantor|1895}}, {{harvtxt|Cantor|1897}}. The English translation is [[#Cantor1955|Cantor 1955]].</ref> The first paper begins by defining set, [[subset]], etc., in ways that would be largely acceptable now. The cardinal and ordinal arithmetic are reviewed. Cantor wanted the second paper to include a proof of the continuum hypothesis, but had to settle for expositing his theory of [[well-ordered set]]s and ordinal numbers. Cantor attempts to prove that if ''A'' and ''B'' are sets with ''A'' [[equinumerous|equivalent]] to a subset of ''B'' and ''B'' equivalent to a subset of ''A'', then ''A'' and ''B'' are equivalent. [[Ernst Schröder (mathematician)|Ernst Schröder]] had stated this theorem a bit earlier, but his proof, as well as Cantor's, was flawed. [[Felix Bernstein (mathematician)|Felix Bernstein]] supplied a correct proof in his 1898 PhD thesis; hence the name [[Cantor–Bernstein–Schröder theorem]].

====One-to-one correspondence====
{{Main|Bijection}}
[[File:Bijection.svg|thumb|A bijective function]]
Cantor's 1874 [[Crelle's Journal|Crelle]] paper was the first to invoke the notion of a [[Bijection|1-to-1 correspondence]], though he did not use that phrase. He then began looking for a 1-to-1 correspondence between the points of the [[unit square]] and the points of a unit [[line segment]]. In an 1877 letter to Richard Dedekind, Cantor proved a far [[Mathematical jargon#stronger|stronger]] result: for any positive integer ''n'', there exists a 1-to-1 correspondence between the points on the unit line segment and all of the points in an [[n-dimensional space|''n''-dimensional space]]. About this discovery Cantor wrote to Dedekind: "{{lang|fr|Je le vois, mais je ne le crois pas!}}" ("I see it, but I don't believe it!")<ref>{{Cite book |last=Wallace |first=David Foster |year=2003|title=Everything and More: A Compact History of Infinity|place=New York|publisher=W.&nbsp;W. Norton and Company|isbn=978-0-393-00338-3|page=[https://archive.org/details/everythingmore00davi/page/259 259]|url=https://archive.org/details/everythingmore00davi/page/259}}</ref> The result that he found so astonishing has implications for geometry and the notion of [[dimension]].

In 1878, Cantor submitted another paper to Crelle's Journal, in which he defined precisely the concept of a 1-to-1 correspondence and introduced the notion of "[[cardinality|power]]" (a term he took from [[Jakob Steiner]]) or "equivalence" of sets: two sets are equivalent (have the same power) if there exists a 1-to-1 correspondence between them. Cantor defined [[countable set]]s (or denumerable sets) as sets which can be put into a 1-to-1 correspondence with the [[natural number]]s, and proved that the rational numbers are denumerable. He also proved that ''n''-dimensional [[Euclidean space]] '''R'''<sup>''n''</sup> has the same power as the [[real number]]s '''R''', as does a countably infinite [[Cartesian product|product]] of copies of '''R'''. While he made free use of countability as a concept, he did not write the word "countable" until 1883. Cantor also discussed his thinking about [[dimension]], stressing that his [[Map (mathematics)|mapping]] between the [[unit interval]] and the unit square was not a [[continuous function|continuous]] one.

This paper displeased Kronecker and Cantor wanted to withdraw it; however, Dedekind persuaded him not to do so and [[Karl Weierstrass]] supported its publication.<ref>[[#Dauben1979|Dauben 1979]], pp. 69, 324 ''63n''. The paper had been submitted in July 1877. Dedekind supported it, but delayed its publication due to Kronecker's opposition. Weierstrass actively supported it.</ref> Nevertheless, Cantor never again submitted anything to Crelle.

====Continuum hypothesis====
{{Main|Continuum hypothesis}}
Cantor was the first to formulate what later came to be known as the [[continuum hypothesis]] or CH: there exists no set whose power is greater than that of the naturals and less than that of the reals (or equivalently, the cardinality of the reals is ''exactly'' aleph-one, rather than just ''at least'' aleph-one). Cantor believed the continuum hypothesis to be true and tried for many years to [[mathematical proof|prove]] it, in vain. His inability to prove the continuum hypothesis caused him considerable anxiety.<ref name="daub280" />

The difficulty Cantor had in proving the continuum hypothesis has been underscored by later developments in the field of mathematics: a 1940 result by [[Kurt Gödel]] and a 1963 one by [[Paul Cohen (mathematician)|Paul Cohen]] together imply that the continuum hypothesis can be neither proved nor disproved using standard [[Zermelo–Fraenkel set theory]] plus the [[axiom of choice]] (the combination referred to as "[[ZFC]]").<ref>Some mathematicians consider these results to have settled the issue, and, at most, allow that it is possible to examine the formal consequences of CH or of its negation, or of axioms that imply one of those. Others continue to look for "natural" or "plausible" axioms that, when added to ZFC, will permit either a proof or refutation of CH, or even for direct evidence for or against CH itself; among the most prominent of these is [[W. Hugh Woodin]]. One of Gödel's last papers argues that the CH is false, and the continuum has cardinality Aleph-2.</ref>

====Absolute infinite, well-ordering theorem, and paradoxes====
In 1883, Cantor divided the infinite into the transfinite and the [[Absolute infinite|absolute]].<ref>{{harvnb|Cantor|1883|pp=587–588}}; English translation: [[#Ewald|Ewald 1996]], pp. 916&ndash;917.</ref>

The transfinite is increasable in magnitude, while the absolute is unincreasable. For example, an ordinal α is transfinite because it can be increased to α&nbsp;+&nbsp;1. On the other hand, the ordinals form an absolutely infinite sequence that cannot be increased in magnitude because there are no larger ordinals to add to it.<ref>[[#Hallett|Hallett 1986]], pp. 41–42.</ref> In 1883, Cantor also introduced the [[Well-ordering theorem|well-ordering principle]] "every set can be well-ordered" and stated that it is a "law of thought".<ref>[[#Moore1982|Moore 1982]], p. 42.</ref>

Cantor extended his work on the absolute infinite by using it in a proof. Around 1895, he began to regard his well-ordering principle as a theorem and attempted to prove it. In 1899, he sent Dedekind a proof of the equivalent aleph theorem: the cardinality of every infinite set is an [[aleph number|aleph]].<ref>[[#Moore1982|Moore 1982]], p. 51. Proof of equivalence: If a set is well-ordered, then its cardinality is an aleph since the alephs are the cardinals of well-ordered sets. If a set's cardinality is an aleph, then it can be well-ordered since there is a one-to-one correspondence between it and the well-ordered set defining the aleph.</ref> First, he defined two types of multiplicities: consistent multiplicities (sets) and inconsistent multiplicities (absolutely infinite multiplicities). Next he assumed that the ordinals form a set, proved that this leads to a contradiction, and concluded that the ordinals form an inconsistent multiplicity. He used this inconsistent multiplicity to prove the aleph theorem.<ref>[[#Hallett|Hallett 1986]], pp. 166–169.</ref> In 1932, Zermelo criticized the construction in Cantor's proof.<ref>Cantor's proof, which is a [[proof by contradiction]], starts by assuming there is a set ''S'' whose cardinality is not an aleph. A function from the ordinals to ''S'' is constructed by successively choosing different elements of ''S'' for each ordinal. If this construction runs out of elements, then the function well-orders the set ''S''. This implies that the cardinality of ''S'' is an aleph, contradicting the assumption about ''S''. Therefore, the function maps all the ordinals one-to-one into ''S''. The function's [[Image (mathematics)|image]] is an inconsistent submultiplicity contained in ''S'', so the set ''S'' is an inconsistent multiplicity, which is a contradiction. Zermelo criticized Cantor's construction: "the intuition of time is applied here to a process that goes beyond all intuition, and a fictitious entity is posited of which it is assumed that it could make ''successive'' arbitrary choices." ([[#Hallett|Hallett 1986]], pp. 169–170.)</ref>

Cantor avoided [[paradox]]es by recognizing that there are two types of multiplicities. In his set theory, when it is assumed that the ordinals form a set, the resulting contradiction implies only that the ordinals form an inconsistent multiplicity. In contrast, [[Bertrand Russell]] treated all collections as sets, which leads to paradoxes. In Russell's set theory, the ordinals form a set, so the resulting contradiction implies that the theory is [[inconsistent]]. From 1901 to 1903, Russell discovered three paradoxes implying that his set theory is inconsistent: the [[Burali-Forti paradox]] (which was just mentioned), [[Cantor's paradox]], and [[Russell's paradox]].<ref>[[#Moore1988|Moore 1988]], pp. 52–53; [[#Moore1981|Moore and Garciadiego 1981]], pp. 330–331.</ref> Russell named paradoxes after [[Cesare Burali-Forti]] and Cantor even though neither of them believed that they had found paradoxes.<ref>[[#Moore1981|Moore and Garciadiego 1981]], pp. 331, 343; [[#Purkert1989|Purkert 1989]], p. 56.</ref>

In 1908, Zermelo published [[Zermelo set theory|his axiom system for set theory]]. He had two motivations for developing the axiom system: eliminating the paradoxes and securing his proof of the [[well-ordering theorem]].<ref>[[#Moore1982|Moore 1982]], pp. 158–160. Moore argues that the latter was his primary motivation.</ref> Zermelo had proved this theorem in 1904 using the [[axiom of choice]], but his proof was criticized for a variety of reasons.<ref>Moore devotes a chapter to this criticism: "Zermelo and His Critics (1904–1908)", [[#Moore1982|Moore 1982]], pp. 85–141.</ref> His response to the criticism included his axiom system and a new proof of the well-ordering theorem. His axioms support this new proof, and they eliminate the paradoxes by restricting the formation of sets.<ref>[[#Moore1982|Moore 1982]], pp. 158–160. [[#Zermelo1908|Zermelo 1908]], pp. 263–264; English translation: [[#Heijenoort|van Heijenoort 1967]], p. 202.</ref>

In 1923, [[John von Neumann]] developed an axiom system that eliminates the paradoxes by using an approach similar to Cantor's—namely, by identifying collections that are not sets and treating them differently. Von Neumann stated that a [[Class (set theory)|class]] is too big to be a set if it can be put into one-to-one correspondence with the class of all sets. He defined a set as a class that is a member of some class and stated the axiom: A class is not a set if and only if there is a one-to-one correspondence between it and the class of all sets. This axiom implies that these big classes are not sets, which eliminates the paradoxes since they cannot be members of any class.<ref>[[#Hallett|Hallett 1986]], pp. 288, 290–291. Cantor had pointed out that inconsistent multiplicities face the same restriction: they cannot be members of any multiplicity. ([[#Hallett|Hallett 1986]], p. 286.)</ref> Von Neumann also used his axiom to prove the well-ordering theorem: Like Cantor, he assumed that the ordinals form a set. The resulting contradiction implies that the class of all ordinals is not a set. Then his axiom provides a one-to-one correspondence between this class and the class of all sets. This correspondence well-orders the class of all sets, which implies the well-ordering theorem.<ref>[[#Hallett|Hallett 1986]], pp. 291–292.</ref> In 1930, Zermelo defined [[Zermelo's models and the axiom of limitation of size|models of set theory that satisfy von Neumann's axiom]].<ref>[[#Zermelo1930|Zermelo 1930]]; English translation: [[#Ewald|Ewald 1996]], pp. 1208–1233.</ref>