articles on second order arithmetic and reverse mathematics. Set theories 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) Kripke–Platek set theory; KP; Pocket set theory General set theory, GST Constructive set theory, CZF Mac Lane set theory and Elementary topos theory Zermelo set theory; Z Zermelo–Fraenkel set theory; ZF, ZFC; Von Neumann–Bernays–Gödel set theory; NBG; (finitely axiomatizable) Ackermann set theory; Scott–Potter set theory New Foundations; NF (finitely axiomatizable) Positive set theory Morse–Kelley set theory; MK; Tarski–Grothendieck set theory; TG; 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 See also References Further reading First-order theories, List of First-order theories, List of First-order theories