List of First-order Theories - Set Theories

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:

1. Use first-order logic with two types.
2. Use ordinary first-order logic, but add a new unary predicate "Set", where "Set(t)" means informally "t is a set".
3. 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