Set theory with a universal set: exploring an untyped universe
Set theory is concerned with the foundations of mathematics. In the original formulations, there were paradoxes concerning the idea of the "set of all sets." Current standard theory (Zermelo-Fraenkel) avoids these paradoxes by restricting the way sets may be formed by other sets specifically to disallow the possibility of forming the set of all sets. In the 1930s, Quine proposed a different form of set theory in which the set of all sets-- the universal set-- is allowed, but other restrictions are placed on these axioms. Since then, the steady interest expressed in these non-standard set theories has been boosted by their relevance to computer science. This text concentrates heavily on Quine's New Foundations, reflecting the author's belief that it provides the richest and most mysterious of the various systems dealing with set theories with a universal set. The result is a work that provides a useful introduction for those new to this topic, and a valuable reference for those already involved in the area.
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NF and related systems
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aleph assertion aussonderung automorphism AxCount axiom of counting axiom of foundation axiom of infinity axiom scheme bijection Boffa Burali-Forti paradox Cantor's theorem complement comprehension axiom consider consistency proofs construction countable defined definition disjoint elementarily equivalent end-extension equiconsistent equivalent existence finite sets Forster free variables function Henson Hinnion homogeneous induction integers invariant isomorphism Journal of Symbolic least model of NF NF is consistent notation obvious order-type ordinals Orey paradox permutation model Petry positive set theory power set predicate proper classes proposition provable prove quantifier Quine atoms recursion relational types REMARK restricted result satisfies self-membered set abstracts set theory setlike permutation singleton Specker's stratified formulae stratified sentences stratimorphic strongly cantorian set subset Suppose Symbolic Logic symmetric term model things transitive sets type theory universal set unstratified well-founded extensional relation well-founded sets well-ordering Wins Gx