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
Famous quotes containing the words set and/or theories:
“I set forth notions that are human and my own, simply as human notions considered in themselves, not as determined and decreed by heavenly ordinance.”
—Michel de Montaigne (1533–1592)
“The two most far-reaching critical theories at the beginning of the latest phase of industrial society were those of Marx and Freud. Marx showed the moving powers and the conflicts in the social-historical process. Freud aimed at the critical uncovering of the inner conflicts. Both worked for the liberation of man, even though Marx’s concept was more comprehensive and less time-bound than Freud’s.”
—Erich Fromm (1900–1980)