A ∈ A;
(ii) if a ∈ A and X ⊆ A is Δ0 on A, then X ∩ a ∈ A;
(iii) if a ∈ A, X ⊆ A is Δ0 on A, and ∀x∈a∃y(<x,y> ∈X), then, for some b ∈ A, ∀x∈a∃y∈b(< x,y> ∈ X).
| This is a file in the archives of the Stanford Encyclopedia of Philosophy. |
(i) if a, b ∈ A, then {a, b} ∈ A andCondition (ii) -- the Δ0-separation scheme -- is a restricted version of Zermelo's axiom of separation. Condition (iii) -- a similarly weakened version of the axiom of replacement -- may be called the Δ0-replacement scheme.(ii) if a ∈ A and X ⊆ A is Δ0 on A, then X ∩ a ∈ A;
(iii) if a ∈ A, X ⊆ A is Δ0 on A, and ∀x∈a∃y(<x,y> ∈X), then, for some b ∈ A, ∀x∈a∃y∈b(< x,y> ∈ X).
It is quite easy to see that if A is a transitive set such that <A, ∈| A> is a model of ZFC, then A is admissible. More generally, the result continues to hold when the power set axiom is omitted from ZFC, so that both H(ω) and H(ω1) are admissible. However, since the latter is uncountable, the Barwise compactness theorem fails to apply to it.
| John L. Bell jbell@uwo.ca |