HC of an Admissible Set

If A is an admissible set, let HC(A) = {x I x E A and x is hereditarily countable in A}. Then HC(A) is admissible. Corollaries are drawn characterizing the "real parts" of admissible sets and the analytical consequences of admissible set theory. ?1. A transitive set A is admissible if it satisfies pairing, set union, J0-separation and J0-bounding (Platek [7], Barwise [1]). An easy theorem of ZFC (ZermeloFraenkel with choice) is that HC, the collection of hereditarily countable sets, is admissible. It is a question of G. Sacks whether this fact is derivable from the axioms for admissible sets alone. In this paper we answer this question affirmatively and use this result to identify the sentences in the language of analysis which hold in all admissible sets (satisfying the axiom of infinity). THEOREM 1. Let A be admissible and HC(A) = {x e A I V transitive closure (x) is countable}. Then HC(A) is admissible. In fact, HC(A) is a C1-elementary submodel of A(HC(A) -< 11A). PROOF. First we establish the admissibility of HC(A). That is, we wish to show the following in A: (*) dVn 3x e HC qp(n, x) -3f fE HC Vn q(n, f(n)) for each J0-formula q (with parameter from HC(A)). We can assume that Vn Vx(qp(n, x) -* x is transitive). Now any transitive x e HC(A) is "coded" by some R e A, R c E in the sense that is isomorphic to e RI>. Conversely, if R e A, R C C is wellfounded, i.e., e R}> is a wellfounded partial ordering, then R codes a transitive x e HC(A). Thus (*) can be transformed into the equivalent: Vin 3R (R wellfounded A 3 T Vm P (R(m), T(m), n)) + 3 R 3T Vn((R)" wellfounded A Vm P((R)" (m), (Tj) (m), n)) for an appropriate recursive P (with parameter from 2w n A). For each n, consider the tree Tn whose nodes are of the from (r, t; 1) where (1) r and t are finite strings of 0's and l's, (2) Vm < lh(t)PQ((m), i(m), n), Received September 27, 1977. 'The research for this paper was supported by NSF Grant ; MCS 76-07033. This paper accompanies a talk delivered by the author at the December, 1977 meeting of the Association for Symbolic Logic, Washington, D. C. The author wishes to thank the referee for his pointed and valuable suggestions.