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). Related definitions 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 L is an admissible set. L is called an amenable set if it is a standard model of KP set theory without the axiom of Δ0-collection. Theorems Admissible sets 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. Cartesian products exist Theorem: If A and B are sets, then there is a set A×B which consists of all ordered pairs (a, b) of elements a of A and b of B. Proof: The set {a} (which is the same as {a, a} by the axiom of extensionality) and the set {a, b} 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 (a, b) is: Thus a superset of A×{b} = {(a, b) | a ∈ A} exists by the axiom of collection. Denote the formula for p above by . Then the following formula is also Δ0 Thus A×{b} itself exists by the axiom of separation. If v is intended to stand for A×{b}, then a Δ0 formula expressing that is: Thus a superset of {A×{b} | b ∈ B} exists by the axiom of collection. Putting in front of that last formula and we get from the axiom of separation that the set {A×{b} | b ∈ B} itself exists. Finally, A×B = {A×{b} | b ∈ B} exists by the axiom of union. Q.E.D. Metalogic The consistency strength of KPω is given by