Axiom of extensionality: Two sets are the same if and only if they have the same elements.
Axiom of induction: φ being a formula, if for all sets x the assumption that φ holds for all elements y of x entails that φ holds, then φ holds for all sets x.
Axiom of pairing: If x, y are sets, then so is, a set containing x and y as its only elements.
Axiom of union: For any set x, there is a set y such that the elements of y are precisely the elements of the elements of x.
Axiom of Σ0-separation: Given any set and any Σ0-formula φ, there is a subset of the original set containing precisely those elements x for which φ holds.
Axiom of Σ0-collection: Given any Σ0-formula φ, if for every set xthere exists a unique set y such that φ holds, then for all sets u there exists a set v such that for every x in u there is a y in v such that φ holds.
Here, a Σ0, or Π0, or Δ0 formula is one all of whose quantifiers are bounded. This means any quantification is the form or
Some but not all authors include an axiom of infinity.
These axioms are weaker than ZFC as they exclude the axioms of powerset, 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. Not adopting Regularity or the Axiom of Choice, KP can be studied as a constructive set theory by dropping the law of excluded middle, without changing any axioms.
Theorem: If A and B are sets, then there is a set A×B which consists of all ordered pairs of elements a of A and b of B. Proof: The set and the set both exist by the axiom of pairing. Thus exists by the axiom of pairing as well. A possible Δ0 formula expressing that p stands for is: Thus a superset of A× = exists by the axiom of collection. Denote the formula for p above by. Then the following formula is also Δ0 Thus A× itself exists by the axiom of separation. If v is intended to stand for A×, then a Δ0 formula expressing that is: Thus a superset of exists by the axiom of collection. Putting in front of that last formula and we get from the axiom of separation that the set itself exists. Finally, A×B = exists by the axiom of union. QED
Admissible sets
A set is called admissible if it is transitive and is a model of Kripke–Platek set theory. An ordinal numberα is called an admissible ordinal if Lα is an admissible set. 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 mapping from γ onto α. If M is a standard model of KP, then the set of ordinals in M is an admissible ordinal. If Lα is a standard model of KP set theory without the axiom of Σ0-collection, then it is said to be an "amenable set".
