Крипке–Платек ақсиомалық жиын теориясы

System of mathematical set theory

The Kripke–Platek set theory (KP), pronounced '/k//r//ɪ//p//k//i// /'/p//l//ɑː//t//ɛ//k/, is an axiomatic set theory developed by Saul Kripke and Richard Platek. The theory can be thought of as roughly the predicative part of ZFC and is considerably weaker than it.

If any set is postulated to exist, such as in the axiom of infinity, then the axiom of empty set is redundant because it is equal to the subset Furthermore, the existence of a member in the universe of discourse, i. e., ∃x(x=x), is implied in certain formulations of first order logic, in which case the axiom of empty set follows from the axiom of Δ0 separation, and is thus redundant.

As noted, the above are weaker than ZFC as they exclude the power set axiom, choice, and sometimes infinity. Also the axioms of separation and collection here are weaker than the corresponding axioms in ZFC because the formulas φ used in these are limited to bounded quantifiers only. The axiom of induction in the context of KP is stronger than the usual axiom of regularity, which amounts to applying induction to the complement of a set (the class of all sets not in the given set).

A set is called admissible if it is transitive and is a model of Kripke–Platek set theory. An ordinal number is called an admissible ordinal if is an admissible set. is called an amenable set if it is a standard model of KP set theory without the axiom of Δ0 collection.

The ordinal α is an admissible ordinal if and only if α is a limit ordinal and there does not exist a γ < α for which there is a Σ1(Lα) mapping from γ onto α. If M is a standard model of KP, then the set of ordinals in M is an admissible ordinal.

The consistency strength of KPω is given by the Bachmann–Howard ordinal. KP fails to prove some common theorems in set theory, such as the Mostowski collapse lemma.