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==History== === Early history === [[File:Arbor porphyrii (from Purchotius' Institutiones philosophicae I, 1730).png|thumb|236x236px|[[Porphyrian tree]] by [[:File:Arbor porphyrii (from Purchotius' Institutiones philosophicae I, 1730).png|Purchotius]] (1730), presenting [[Aristotle]]'s [[Categories (Aristotle)|Categories]].]] The basic notion of grouping objects has existed since at least the [[Natural number#History|emergence of numbers]], and the notion of treating sets as their own objects has existed since at least the [[Tree of Porphyry]], 3rd-century AD. The simplicity and ubiquity of sets makes it hard to determine the origin of sets as now used in mathematics, however, [[Bernard Bolzano]]'s ''[[Paradoxes of the Infinite]]'' (''Paradoxien des Unendlichen'', 1851) is generally considered the first rigorous introduction of sets to mathematics. In his work, he (among other things) expanded on [[Galileo's paradox]], and introduced [[one-to-one correspondence]] of infinite sets, for example between the [[Interval (mathematics)|intervals]] <math>[0,5]</math> and <math>[0,12]</math> by the relation <math>5y = 12x</math>. However, he resisted saying these sets were [[equinumerous]], and his work is generally considered to have been uninfluential in mathematics of his time.<ref>{{Citation |last=Ferreirós |first=José |title=The Early Development of Set Theory |date=2024 |encyclopedia=The Stanford Encyclopedia of Philosophy |editor-last=Zalta |editor-first=Edward N. |url=https://plato.stanford.edu/entries/settheory-early/ |access-date=2025-01-04 |edition=Winter 2024 |publisher=Metaphysics Research Lab, Stanford University |editor2-last=Nodelman |editor2-first=Uri |archive-date=2023-03-20 |archive-url=https://archive.today/20230320205811/https://plato.stanford.edu/entries/settheory-early/ |url-status=live }}</ref><ref>{{Citation |last=Bolzano |first=Bernard |title=Einleitung zur Größenlehre und erste Begriffe der allgemeinen Größenlehre |volume=II, A, 7 |page=152 |year=1975 |editor-last=Berg |editor-first=Jan |series=Bernard-Bolzano-Gesamtausgabe, edited by Eduard Winter et al. |location=Stuttgart, Bad Cannstatt |publisher=Friedrich Frommann Verlag |isbn=3-7728-0466-7 |author-link=Bernard Bolzano}}</ref> Before mathematical set theory, basic concepts of [[infinity]] were considered to be solidly in the domain of philosophy (see: ''[[Infinity (philosophy)]]'' and ''{{Section link|Infinity|History}}''). Since the 5th century BC, beginning with Greek philosopher [[Zeno of Elea]] in the West (and early [[Indian mathematics|Indian mathematicians]] in the East), mathematicians had struggled with the concept of infinity. With the [[History of calculus|development of calculus]] in the late 17th century, philosophers began to generally distinguish between [[Actual infinity|actual and potential infinity]], wherein mathematics was only considered in the latter.<ref>{{Citation |last=Zenkin |first=Alexander |title=Logic Of Actual Infinity And G. Cantor's Diagonal Proof Of The Uncountability Of The Continuum |periodical=The Review of Modern Logic |volume=9 |issue=30 |pages=27–80 |year=2004 |url=http://projecteuclid.org/euclid.rml/1203431978 |access-date=2025-01-04 |archive-date=2020-09-22 |archive-url=https://web.archive.org/web/20200922022622/https://projecteuclid.org/euclid.rml/1203431978 |url-status=live }}</ref> [[Carl Friedrich Gauss]] famously stated: "Infinity is nothing more than a figure of speech which helps us talk about limits. The notion of a completed infinity doesn't belong in mathematics."<ref>{{cite book |last1=Dunham |first1=William |url=https://archive.org/details/journeythroughge00dunh_359 |title=Journey through Genius: The Great Theorems of Mathematics |publisher=Penguin |year=1991 |isbn=9780140147391 |page=[https://archive.org/details/journeythroughge00dunh_359/page/n267 254] |url-access=limited}}</ref> Development of mathematical set theory was motivated by several mathematicians. [[Bernhard Riemann]]'s lecture ''On the Hypotheses which lie at the Foundations of Geometry'' (1854) proposed new ideas about [[topology]], and about basing mathematics (especially geometry) in terms of sets or [[manifold]]s in the sense of a [[Class (set theory)|class]] (which he called ''Mannigfaltigkeit'') now called [[point-set topology]]. The lecture was published by [[Richard Dedekind]] in 1868, along with Riemann's paper on [[trigonometric series]] (which presented the [[Riemann integral]]), The latter was a starting point a movement in [[real analysis]] for the study of “seriously” [[discontinuous function]]s. A young [[Georg Cantor]] entered into this area, which led him to the study of [[Point set|point-sets]]. Around 1871, influenced by Riemann, Dedekind began working with sets in his publications, which dealt very clearly and precisely with [[equivalence relations]], [[Partition of a set|partitions of sets]], and [[homomorphisms]]. Thus, many of the usual set-theoretic procedures of twentieth-century mathematics go back to his work. However, he did not publish a formal explanation of his set theory until 1888. === Naive set theory === {{Main|Naive set theory}} [[File:Georg Cantor 1894.jpg|thumb|160px|[[Georg Cantor]], 1894]] Set theory, as understood by modern mathematicians, is generally considered to be founded by a single paper in 1874 by [[Georg Cantor]] titled ''[[On a Property of the Collection of All Real Algebraic Numbers]]''.<ref name="cantor1874">{{citation|last=Cantor|first=Georg|author-link=Georg Cantor|year=1874|title=Ueber eine Eigenschaft des Inbegriffes aller reellen algebraischen Zahlen|url=http://www.digizeitschriften.de/main/dms/img/?PPN=GDZPPN002155583|journal=[[Journal für die reine und angewandte Mathematik]]|language=de|volume=1874|issue=77|pages=258–262|doi=10.1515/crll.1874.77.258|s2cid=199545885|access-date=2013-01-31|archive-date=2012-06-04|archive-url=https://archive.today/20120604145721/http://www.digizeitschriften.de/main/dms/img/?PPN=GDZPPN002155583|url-status=live}}</ref><ref>{{citation |first=Philip |last=Johnson |year=1972 |title=A History of Set Theory |publisher=Prindle, Weber & Schmidt |isbn=0-87150-154-6 |url-access=registration |url=https://archive.org/details/historyofsettheo0000unse }}</ref><ref>{{Citation |last=Dauben |first=Joseph |title=Georg Cantor: His Mathematics and Philosophy of the Infinite |pages=30–54 |year=1979 |publisher=Harvard University Press |isbn=0-674-34871-0 |author-link=Joseph Dauben}}.</ref> In his paper, he developed the notion of [[cardinality]], comparing the sizes of two sets by setting them in one-to-one correspondence. His "revolutionary discovery" was that the set of all [[real number]]s is [[Uncountable set|uncountable]], that is, one cannot put all real numbers in a list. This theorem is proved using [[Cantor's first set theory article#The proofs|Cantor's first uncountability proof]], which differs from the more familiar proof using his [[Cantor's diagonal argument|diagonal argument]]. Cantor introduced fundamental constructions in set theory, such as the [[power set]] of a set ''A'', which is the set of all possible [[subset]]s of ''A''. He later proved that the size of the power set of ''A'' is strictly larger than the size of ''A'', even when ''A'' is an infinite set; this result soon became known as [[Cantor's theorem]]. Cantor developed a theory of [[transfinite numbers]], called [[Cardinal number|cardinals]] and [[Ordinal number|ordinals]], which extended the arithmetic of the natural numbers. His notation for the cardinal numbers was the Hebrew letter <math>\aleph</math> ([[ℵ]], [[aleph]]) with a natural number subscript; for the ordinals he employed the Greek letter <math>\omega</math> ({{script|Grek|ω}}, [[omega]]). Set theory was beginning to become an essential ingredient of the new “modern” approach to mathematics. Originally, Cantor's theory of transfinite numbers was regarded as counter-intuitive – even shocking. This caused it to encounter resistance from mathematical contemporaries such as [[Leopold Kronecker]] and [[Henri Poincaré]] and later from [[Hermann Weyl]] and [[L. E. J. Brouwer]], while [[Ludwig Wittgenstein]] raised [[Philosophical objections to Cantor's theory|philosophical objections]] (see: ''[[Controversy over Cantor's theory]]'').{{Efn|The objections to Cantor's work were occasionally fierce: Leopold Kronecker's public opposition and personal attacks included describing Cantor as a "scientific charlatan", a "renegade" and a "corrupter of youth". Kronecker objected to Cantor's proofs that the algebraic numbers are countable, and that the transcendental numbers are uncountable, results now included in a standard mathematics curriculum. Writing decades after Cantor's death, Wittgenstein lamented that mathematics is "ridden through and through with the pernicious idioms of set theory", which he dismissed as "utter nonsense" that is "laughable" and "wrong".}} Dedekind's algebraic style only began to find followers in the 1890s [[File:Young frege.jpg|left|thumb|209x209px|[[Gottlob Frege]], c. 1879]] Despite the controversy, Cantor's set theory gained remarkable ground around the turn of the 20th century with the work of several notable mathematicians and philosophers. Richard Dedekind, around the same time, began working with sets in his publications, and famously constructing the real numbers using [[Dedekind cuts]]. He also worked with [[Giuseppe Peano]] in developing the [[Peano axioms]], which formalized natural-number arithmetic, using set-theoretic ideas, which also introduced the [[epsilon]] symbol for [[Element (mathematics)|set membership]]. Possibly most prominently, [[Gottlob Frege]] began to develop his ''[[Foundations of Arithmetic]]''. In his work, Frege tries to ground all mathematics in terms of logical axioms using Cantor's cardinality. For example, the sentence "the number of horses in the barn is four" means that four objects fall under the concept ''horse in the barn''. Frege attempted to explain our grasp of numbers through cardinality ('the number of...', or <math> Nx: Fx </math>), relying on [[Hume's principle]]. [[File:Bertrand Russell photo (cropped).jpg|thumb|203x203px|[[Bertrand Russell]], 1936.]] However, Frege's work was short-lived, as it was found by [[Bertrand Russell]] that his axioms lead to a [[contradiction]]. Specifically, Frege's [[Basic Law V]] (now known as the [[axiom schema of unrestricted comprehension]]). According to [[Basic Law V]], for any sufficiently well-defined [[Property (philosophy)|property]], there is the set of all and only the objects that have that property. The contradiction, called [[Russell's paradox]], is shown as follows: Let ''R'' be the set of all sets that are not members of themselves. (This set is sometimes called "the Russell set".) If ''R'' is not a member of itself, then its definition entails that it is a member of itself; yet, if it is a member of itself, then it is not a member of itself, since it is the set of all sets that are not members of themselves. The resulting contradiction is Russell's paradox. In symbols: : <math>\text{Let } R = \{ x \mid x \not \in x \} \text{, then } R \in R \iff R \not \in R</math> This came around a time of several [[paradox]]es or counter-intuitive results. For example, that the [[parallel postulate]] cannot be proved, the existence of [[mathematical object]]s that cannot be computed or explicitly described, and the existence of theorems of arithmetic that cannot be proved with [[Peano arithmetic]]. The result was a [[foundational crisis of mathematics]].
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