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)
- 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
Read more about this topic: List Of First-order Theories
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