Editing Russell's paradox (section)

== Set-theoretic responses ==
From the [[principle of explosion]] of [[classical logic]], ''any'' proposition can be proved from a [[contradiction]].  Therefore, the presence of contradictions like Russell's paradox in an axiomatic set theory is disastrous; since if any formula can be proved true it destroys the conventional meaning of truth and falsity.  Further, since set theory was seen as the basis for an axiomatic development of all other branches of mathematics, Russell's paradox threatened the foundations of mathematics as a whole. This motivated a great deal of research around the turn of the 20th century to develop a consistent (contradiction-free) set theory.

In 1908, [[Ernst Zermelo]] proposed an [[axiomatic system|axiomatization]] of set theory that avoided the paradoxes of naive set theory by replacing arbitrary set comprehension with weaker existence axioms, such as his [[axiom of separation]] (''Aussonderung''). (Avoiding paradox was not Zermelo's original intention, but instead to document which assumptions he used in proving the [[well-ordering theorem]].)<ref name="Maddy">P. Maddy, "[https://www.cs.umd.edu/~gasarch/BLOGPAPERS/belaxioms1.pdf Believing the Axioms I]" (1988). Association for Symbolic Logic.</ref> Modifications to this axiomatic theory proposed in the 1920s by [[Abraham Fraenkel]], [[Thoralf Skolem]], and by Zermelo himself resulted in the axiomatic set theory called [[ZFC]]. This theory became widely accepted once Zermelo's [[axiom of choice]] ceased to be controversial, and ZFC has remained the canonical [[axiomatic set theory]] down to the present day.

ZFC does not assume that, for every property, there is a set of all things satisfying that property. Rather, it asserts that given any set ''X'', any subset of ''X'' definable using [[first-order logic]] exists. The object ''R'' defined by Russell's paradox above cannot be constructed as a subset of any set ''X'', and is therefore not a set in ZFC. In some extensions of ZFC, notably in [[von Neumann–Bernays–Gödel set theory]], objects like ''R'' are called [[proper class]]es.

ZFC is silent about types, although the [[Von Neumann universe|cumulative hierarchy]] has a notion of layers that resemble types. Zermelo himself never accepted Skolem's formulation of ZFC using the language of first-order logic. As José Ferreirós notes, Zermelo insisted instead that "propositional functions (conditions or predicates) used for separating off subsets, as well as the replacement functions, can be 'entirely ''arbitrary''{{'}} [ganz ''beliebig'']"; the modern interpretation given to this statement is that Zermelo wanted to include [[higher-order logic|higher-order quantification]] in order to avoid [[Skolem's paradox]]. Around 1930, Zermelo also introduced (apparently independently of von Neumann), the [[axiom of foundation]], thus—as Ferreirós observes—"by forbidding 'circular' and 'ungrounded' sets, it [ZFC] incorporated one of the crucial motivations of TT [type theory]—the principle of the types of arguments". This 2nd order ZFC preferred by Zermelo, including axiom of foundation, allowed a rich cumulative hierarchy. Ferreirós writes that "Zermelo's 'layers' are essentially the same as the types in the contemporary versions of simple TT [type theory] offered by Gödel and Tarski. One can describe the cumulative hierarchy into which Zermelo developed his models as the universe of a cumulative TT in which transfinite types are allowed. (Once we have adopted an impredicative standpoint, abandoning the idea that classes are constructed, it is not unnatural to accept transfinite types.) Thus, simple TT and ZFC could now be regarded as systems that 'talk' essentially about the same intended objects. The main difference is that TT relies on a strong higher-order logic, while Zermelo employed second-order logic, and ZFC can also be given a first-order formulation. The first-order 'description' of the cumulative hierarchy is much weaker, as is shown by the existence of countable models (Skolem's paradox), but it enjoys some important advantages."<ref name="Ferreirós2008">{{cite book|author=José Ferreirós|title=Labyrinth of Thought: A History of Set Theory and Its Role in Modern Mathematics|year=2008|publisher=Springer|isbn=978-3-7643-8350-3|edition=2nd|at=§ Zermelo's cumulative hierarchy pp. 374-378}}</ref>

In ZFC, given a set ''A'', it is possible to define a set ''B'' that consists of exactly the sets in ''A'' that are not members of themselves. ''B'' cannot be in ''A'' by the same reasoning in Russell's Paradox. This variation of Russell's paradox shows that no set contains everything.

Through the work of Zermelo and others, especially [[John von Neumann]], the structure of what some see as the "natural" objects described by ZFC eventually became clear: they are the elements of the [[von Neumann universe]], ''V'', built up from the [[empty set]] by [[transfinite recursion|transfinitely iterating]] the [[power set]] operation. It is thus now possible again to reason about sets in a non-axiomatic fashion without running afoul of Russell's paradox, namely by reasoning about the elements of ''V''. Whether it is ''appropriate'' to think of sets in this way is a point of contention among the rival points of view on the [[philosophy of mathematics]].

Other solutions to Russell's paradox, with an underlying strategy closer to that of [[type theory]], include [[Willard van Orman Quine|Quine]]'s [[New Foundations]] and [[Scott–Potter set theory]]. Yet another approach is to define multiple membership relation with appropriately modified comprehension scheme, as in the [[Double extension set theory]].