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The usual signature of set theory has one binary relation ∈, no constants, and no functions. Some of the theories below are "class theories" which have two sorts of object, sets and classes. There are three common ways of handling this in first-order logic: Use first-order logic with two types. Use ordinary first-order logic, but add a new unary predicate "Set", where "Set(t)" means informally "t is a set". Use ordinary first-order logic, and instead of adding a new predicate to the language, treat "Set(t)" as an abbreviation for "∃y t∈y" Some first order set theories include: Weak theories lacking powersets: S' (Tarski, Mostowski, and Robinson, 1953); (finitely axiomatizable) General set theory; Kripke-Platek set theory; Zermelo set theory; Ackermann set theory Zermelo-Fraenkel set theory; Von Neumann-Bernays-Gödel set theory; (finitely axiomatizable) Morse–Kelley set theory; Tarski–Grothendieck set theory; New Foundations; (finitely axiomatizable) Scott-Potter set theory Positive set theory Some extra first order axioms that can be added to one of these (usually ZF) include: axiom of choice, axiom of dependent choice Generalized continuum hypothesis Martin's axiom (usually together with the negation of the continuum hypothesis), Martin's maximum ◊ and ♣ Axiom of constructibility (V=L) proper forcing axiom analytic determinacy, projective determinacy, Axiom of determinacy Many large cardinal axioms |
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