Editing Set theory (section)

== Controversy ==
{{main|Controversy over Cantor's theory}}

From set theory's inception, some mathematicians have objected to it as a [[foundations of mathematics|foundation for mathematics]]. The most common objection to set theory, one [[Leopold Kronecker|Kronecker]] voiced in set theory's earliest years, starts from the [[mathematical constructivism|constructivist]] view that mathematics is loosely related to computation. If this view is granted, then the treatment of infinite sets, both in [[naive set theory|naive]] and in axiomatic set theory, introduces into mathematics methods and objects that are not computable even in principle. The feasibility of constructivism as a substitute foundation for mathematics was greatly increased by [[Errett Bishop]]'s influential book ''Foundations of Constructive Analysis''.<ref>{{citation|title=Foundations of Constructive Analysis|last=Bishop|first=Errett|publisher=Academic Press|year=1967|isbn=4-87187-714-0|location=New York|author-link=Errett Bishop|url=https://books.google.com/books?id=o2mmAAAAIAAJ}}</ref>

A different objection put forth by [[Henri Poincaré]] is that defining sets using the axiom schemas of [[Axiom schema of specification|specification]] and [[Axiom schema of replacement|replacement]], as well as the [[axiom of power set]], introduces [[impredicativity]], a type of [[Circular definition|circularity]], into the definitions of mathematical objects. The scope of predicatively founded mathematics, while less than that of the commonly accepted Zermelo–Fraenkel theory, is much greater than that of constructive mathematics, to the point that [[Solomon Feferman]] has said that "all of scientifically applicable analysis can be developed [using predicative methods]".<ref>{{citation|title=In the Light of Logic|last=Feferman|first=Solomon|publisher=Oxford University Press|year=1998|isbn=0-195-08030-0|location=New York|pages=280–283, 293–294|author-link=Solomon Feferman|url=https://books.google.com/books?id=1rjnCwAAQBAJ}}</ref>

[[Ludwig Wittgenstein]] condemned set theory philosophically for its connotations of [[mathematical platonism]].<ref>{{Cite SEP|url-id=wittgenstein-mathematics|last=Rodych|first=Victor|date=Jan 31, 2018|title=Wittgenstein's Philosophy of Mathematics|edition=Spring 2018}}</ref> He wrote that "set theory is wrong", since it builds on the "nonsense" of fictitious symbolism, has "pernicious idioms", and that it is nonsensical to talk about "all numbers".<ref>{{citation |last=Wittgenstein |first=Ludwig |year=1975 |title=Philosophical Remarks, §129, §174 |publisher=Oxford: Basil Blackwell |isbn=0-631-19130-5 }}</ref> Wittgenstein identified mathematics with algorithmic human deduction;{{Sfn|Rodych|2018|loc=[https://plato.stanford.edu/entries/wittgenstein-mathematics/#WittInteConsForm §2.1]|ps=: "When we prove a theorem or decide a proposition, we operate in a purely formal, syntactical manner. In doing mathematics, we do not discover pre-existing truths that were 'already there without one knowing' (PG 481)—we invent mathematics, bit-by-little-bit."  Note, however, that Wittgenstein does ''not'' identify such deduction with [[philosophical logic]]; cf. Rodych [https://plato.stanford.edu/entries/wittgenstein-mathematics/#WittMathTrac §1], paras. 7-12.}} the need for a secure foundation for mathematics seemed, to him, nonsensical.{{Sfn|Rodych|2018|loc=[https://plato.stanford.edu/entries/wittgenstein-mathematics/#WittLateCritSetTheoNonEnumVsNonDenu §3.4]|ps=: "Given that mathematics is a '{{small caps|motley}} of techniques of proof' (RFM III, §46), it does not require a foundation (RFM VII, §16) and it cannot be given a self-evident foundation (PR §160; WVC 34 & 62; RFM IV, §3). Since set theory was invented to provide mathematics with a foundation, it is, minimally, unnecessary."}}  Moreover, since human effort is necessarily finite, Wittgenstein's philosophy required an ontological commitment to radical [[Constructivism (math)|constructivism]] and [[finitism]].  Meta-mathematical statements – which, for Wittgenstein, included any statement quantifying over infinite domains, and thus almost all modern set theory – are not mathematics.{{Sfn|Rodych|2018|loc=[https://plato.stanford.edu/entries/wittgenstein-mathematics/#WittInteFini §2.2]|ps=: "An expression quantifying over an infinite domain is never a meaningful proposition, not even when we have proved, for instance, that a particular number {{mvar|n}} has a particular property."}}  Few modern philosophers have adopted Wittgenstein's views after a spectacular blunder in ''[[Remarks on the Foundations of Mathematics]]'': Wittgenstein attempted to refute [[Gödel's incompleteness theorems]] after having only read the abstract. As reviewers [[Georg Kreisel|Kreisel]], [[Paul Bernays|Bernays]], [[Michael Dummett|Dummett]], and [[R. L. Goodstein|Goodstein]] all pointed out, many of his critiques did not apply to the paper in full. Only recently have philosophers such as [[Crispin Wright]] begun to rehabilitate Wittgenstein's arguments.{{Sfn|Rodych|2018|loc=[https://plato.stanford.edu/entries/wittgenstein-mathematics/#WittGodeUndeMathProp §3.6]}}

[[category theory|Category theorists]] have proposed [[topos theory]] as an alternative to traditional axiomatic set theory. Topos theory can interpret various alternatives to that theory, such as [[mathematical constructivism|constructivism]], finite set theory, and [[Turing Machine|computable]] set theory.<ref>{{citation|last1=Ferro|first1=Alfredo|last2=Omodeo|first2=Eugenio G.|last3=Schwartz|first3=Jacob T.|date=September 1980|title=Decision Procedures for Elementary Sublanguages of Set Theory. I. Multi-Level Syllogistic and Some Extensions|journal=[[Communications on Pure and Applied Mathematics]]|volume=33|issue=5|pages=599–608|doi=10.1002/cpa.3160330503}}</ref><ref>{{citation|url=https://archive.org/details/computablesetthe00cant/page/|title=Computable Set Theory|last1=Cantone|first1=Domenico|last2=Ferro|first2=Alfredo|last3=Omodeo|first3=Eugenio G.|publisher=[[Clarendon Press]]|year=1989|isbn=0-198-53807-3|series=International Series of Monographs on Computer Science, Oxford Science Publications|location=Oxford, UK|pages=[https://archive.org/details/computablesetthe00cant/page/ xii, 347]|url-access=registration}}</ref> Topoi also give a natural setting for forcing and discussions of the independence of choice from ZF, as well as providing the framework for [[pointless topology]] and [[Stone space]]s.<ref>{{citation|title=Sheaves in Geometry and Logic: A First Introduction to Topos Theory|last1=Mac Lane|first1=Saunders|last2=Moerdijk|first2=leke|publisher=Springer-Verlag|year=1992|isbn=978-0-387-97710-2|author-link=Saunders Mac Lane|url=https://books.google.com/books?id=SGwwDerbEowC}}</ref>

An active area of research is the [[univalent foundations]] and related to it [[homotopy type theory]]. Within homotopy type theory, a set may be regarded as a homotopy 0-type, with [[universal properties]] of sets arising from the inductive and recursive properties of [[higher inductive type]]s. Principles such as the [[axiom of choice]] and the [[law of the excluded middle]] can be formulated in a manner corresponding to the classical formulation in set theory or perhaps in a spectrum of distinct ways unique to type theory. Some of these principles may be proven to be a consequence of other principles. The variety of formulations of these axiomatic principles allows for a detailed analysis of the formulations required in order to derive various mathematical results.<ref>{{nlab|id=homotopy+type+theory|title=homotopy type theory}}</ref><ref>[http://homotopytypetheory.org/book/ ''Homotopy Type Theory: Univalent Foundations of Mathematics''] {{Webarchive|url=https://web.archive.org/web/20210122181140/http://homotopytypetheory.org/book/ |date=2021-01-22 }}. The Univalent Foundations Program. [[Institute for Advanced Study]].</ref>