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Published online by Cambridge University Press: 12 March 2014
In [1978] Harrington and MacQueen proved that if B is an (A, E)-semirecursive subset of A, such that the functions in BA can be coded as elements of A in an (A, E)-recursive way, then ENV(A, E) is closed under the existential quantifier ∃T ∈ B.
Later Moschovakis showed that if ENV(Vκ, ∈, E) is closed under the quantifier ∃t ∈ λ, where λ is the p-cofinality of κ, then
the p-cofinality of κ is the least ordinal λ for which there exists a (κ, <, E)-recursive partial function ƒ into κ, such that ƒ∣λ is total from λ onto an unbounded subset of κ.
In this paper we prove that for any infinite ordinal κ if p-card(κ) = κ, then ENV(κ, <, E) is closed under ∃t ∈ μ, for μ < p-cf(κ); p-cf(κ) is the “boldface” analog of p-cf((κ) and p-card(κ) is defined similarly.
From this follows that for any infinite ordinal κ the following two statements are equivalent.
(i) ENV(κ, <, E) is closed under bounded existential quantification.
(ii) ENV(κ, <, E) = ENV(κ, <, E#) or p-cf(κ) = κ.
We also show that we cannot omit any of the hypotheses in the above theorem.
We follow mainly the notation of Kechris and Moschovakis [1977].