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A converse of the Barwise completeness theorem

Extract

In this paper a converse of Barwise's completeness theorem is proved by cut-elimination considerations applied to inductive definitions. We show that among the transitive sets T satisfying some weak closure conditions (closure under primitive-recursive set-functions is more than enough), only the unions of admissible sets satisfy Barwise's completeness theorem in the form stating that if φT is a sentence which has a derivation (in the universe) then φ has a derivation in T. See §1 for the origin of the problem in Barwise's paper [Ba].

Stated quite briefly the proof is as follows (a step-by-step account including relevant definitions is given in the body of the paper):

Let T be a transitive prim.-rec. closed set, and let is nonempty, transitive and closed under pairs}. For each let κ(A) be the supremum of closure ordinals of first-order positive operators on subsets of A (first-order with respect to By Theorem 1 of [BGM], it is enough to prove that rank(T) in order to obtain that T is a union of admissible sets. (The rank of a set is defined by rank(x) = sup y ∊ x (rank(y) + 1); since T is prim.-rec. closed, rank(T) = smallest ordinal not in T.)

Let We show how to find in T (in fact, in L ω (A)) a derivable sentence τ that has no derivation D such that rank(D) ≤ α. Thus, if τ is to have a derivation in T, rank(T) > α. α is arbitrary (< κ(A)), so rank(T) ≥ κ(A). Q.E.D.

References

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[Ba] Barwise, J., Infinitary logic and admissible sets, this Journal, vol. 34 (1969), pp. 226252.
[BGM] Barwise, K. J., Gandy, R. O. and Moschovakis, Y. N., The next admissible set, this Journal, vol. 36 (1971), pp. 108120.
[JK] Jensen, R. and Karp, C., Primitive recursive set functions, Proceedings of Symposia in Pure Mathematics, vol. 13, Part I, American Mathematical Society, Providence, R.I., 1971, pp. 143167.
[Ka] Karp, C., An algebraic proof of the Barwise compactness theorem, Lecture Notes in Mathematics, vol. 72, Springer-Verlag, Berlin and New York, 1968, pp. 8095.
[Ke] Keisler, H. J., Model theory for infinitary logic, North-Holland, Amsterdam, 1971.
[Lé] Lévy, A., A hierarchy of formulas in set theory, Memoirs of the American Mathematical Society, No. 57, 1965.
[Ta] Tait, W. W., Normal derivability in classical logic, Lecture Notes in Mathematics, vol. 72, Springer-Verlag, Berlin and New York, 1968, pp. 204236.

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