#### Supplement to Infinitary Logic

## Definition of the Concept of Admissible Set

A nonempty transitive set*A*is said to be

*admissible*when the following conditions are satisfied:

(i) ifCondition (ii) -- the Δa, b∈A, then {a, b} ∈AandA∈A;(ii) if

a∈AandX⊆Ais Δ_{0}onA, thenX∩a∈A;(iii) if

a∈A,X⊆Ais Δ_{0}onA, and ∀x∈a∃y(<x,y> ∈X), then, for someb∈A, ∀x∈a∃y∈b(<x,y> ∈X).

_{0}-

*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 Δ

_{0}-

*replacement scheme*.

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*(ω_{1}) are admissible. However, since the
latter is uncountable, the Barwise compactness theorem fails to apply
to it.