Editing Set theory (section)

=== 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&nbsp;– 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]].