Existentially Incomplete Tame Models and a Conjecture of Ellentuck

Abstract

We construct a recursive ultrapower T / U such that T / U is a tame l-model in the sense of [6, $31 and T / U is existentially incomplete in the models of l l 2 arithmetic. This enables us to answer in the negative a question about closure with respect to recursive fibers of certain special semirings r of isols termed tame models by Barback. Erik Ellentuck had conjuctured that all such semirings enjoy the closure property in quaestion. Our result is that while many do, some do not.Mathematics Subject Classification: 03D50, 02C65, 03F30.We shall use "C" to denote inclusion, "C" to denote proper inclusion. Let I' be a countable subsemiring of the full semiring A of isols, with w C I' where w is the set of the ordinary natural numbers. By AR we denote the class of regressive isols. In 1986 BARBACK introduced in (11 the notion of a "tame model". A characterization of I' being a tame model (this is not BARBACK'S original definition, but equivalent to it) is as follows: I' = {f~(x) : f is a nondecreasing A1 function on w and fi\ is the where X is an infinite regressive is01 with the property that for any A, function g on w we have g A ( x ) is defined and equal to fA(x) for some nondecreasing A1 function f . Each such I' is II2 correct, i.e., it satisfies all the 112 truths of ordinary arithmetic (and, in fact, satisfies all w-true II2 sentences formulated in the stronger language LN discussed, for instance, in [5, $21); moreover, as shown in [Z], each such I' is minimal in this respect, i.e., it has no proper IIz-correct subsemirings other than w .According to BARBACK (in private correspondence), it was conjectured by the late ERIK ELLENTUCK that all tame models enjoy what was referred to in [8] as "Property (P)", which ammounts to the assertion that for every is01 Y E I' andfor some Z E A, then Y = ~A ( Z ' ) for some 2' E J?. In [8] we pointed out that this is equivalent to the conjecture that all tame models are existentially complete. (The only thing missing from [8] that is canonical extension of f to A } ,