Editing Willard Van Orman Quine (section)

===Set theory===
While his contributions to logic include elegant expositions and a number of technical results, it is in [[set theory]] that Quine was most innovative. He always maintained that mathematics required set theory and that set theory was quite distinct from logic. He flirted with [[Nelson Goodman]]'s [[nominalism]] for a while<ref>Nelson Goodman and W. V. O. Quine, [http://www.ditext.com/quine/stcn.html "Steps Toward a Constructive Nominalism"], ''Journal of Symbolic Logic'', 12 (1947): 105–122.</ref> but backed away when he failed to find a nominalist grounding of mathematics.<ref name="SEP-Nom">{{Cite encyclopedia |title=Nominalism in the Philosophy of Mathematics |encyclopedia=The Stanford Encyclopedia of Philosophy |publisher=Metaphysics Research Lab, Stanford University |url=https://plato.stanford.edu/archives/fall2020/entries/nominalism-mathematics/ |last=Bueno |first=Otávio |date=2020 |edition=Fall 2020}}</ref>

Over the course of his career, Quine proposed three axiomatic set theories.
* [[New Foundations]], NF, creates and manipulates sets using a single axiom schema for set admissibility, namely an axiom schema of stratified comprehension, whereby all individuals satisfying a stratified formula compose a set. A stratified formula is one that [[type theory]] would allow, were the [[ontology]] to include types. However, Quine's set theory does not feature types. The metamathematics of NF are curious. NF allows many "large" sets the now-canonical [[ZFC]] set theory does not allow, even sets for which the [[axiom of choice]] does not hold. Since the axiom of choice holds for all finite sets, the failure of this axiom in NF proves that NF includes infinite sets. The consistency of NF relative to other formal systems adequate for mathematics is an open question, albeit that a number of candidate proofs are current in the NF community suggesting that NF is equiconsistent with [[Zermelo set theory]] without Choice. A modification of NF, [[New Foundations|NFU]], due to R. B. Jensen and admitting [[urelement]]s (entities that can be members of sets but that lack elements), turns out to be consistent relative to [[Peano arithmetic]], thus vindicating the intuition behind NF. NF and NFU are the only Quinean set theories with a following. For a derivation of foundational mathematics in NF, see Rosser (1952);
* The set theory of ''Mathematical Logic'' is NF augmented by the [[proper class]]es of [[von Neumann–Bernays–Gödel set theory]], except axiomatized in a much simpler way;
* The set theory of ''Set Theory and Its Logic'' does away with stratification and is almost entirely derived from a single axiom schema. Quine derived the foundations of mathematics once again. This book includes the definitive exposition of Quine's theory of virtual sets and relations, and surveyed axiomatic set theory as it stood circa 1960.

All three set theories admit a universal class, but since they are free of any [[hierarchy]] of [[Type (metaphysics)|types]], they have no need for a distinct universal class at each type level.

Quine's set theory and its background logic were driven by a desire to minimize posits; each innovation is pushed as far as it can be pushed before further innovations are introduced. For Quine, there is but one connective, the [[Sheffer stroke]], and one quantifier, the [[universal quantifier]]. All polyadic [[Predicate (mathematical logic)|predicates]] can be reduced to one dyadic predicate, interpretable as set membership. His rules of proof were limited to [[modus ponens]] and substitution. He preferred [[logical conjunction|conjunction]] to either [[disjunction]] or the [[Material conditional|conditional]], because conjunction has the least semantic ambiguity. He was delighted to discover early in his career that all of first order logic and set theory could be grounded in a mere two primitive notions: [[set abstraction|abstraction]] and [[inclusion (set theory)|inclusion]]. For an elegant introduction to the parsimony of Quine's approach to logic, see his "New Foundations for Mathematical Logic", ch.&nbsp;5 in his ''From a Logical Point of View''.