Sub-arithmetical ultrapowers: a survey

Recursive ultrapowers, simple models, and cofinal extensions

“…3 we shall have occasion to make a slight refinement in the specification of F given in [21, and also note that so long as our uniform enumeration of Z ~ sets has a certain standard property, we can conclude that I has recursive domain (and hence can be extended to a total recursive function). The following relationship was alluded to in [2] and explicitly stated in [4]; we shall provide a proof, since we are unaware of any handy reference for one.…”

“…We shall give proofs only for the case ofA ~ and ~(A~ but these proofs relativize without difficulty to A~ 3(A o), noting that in the instance of Theorem 4.1 one needs a relativized version of Lemma 2.2 to replace our use of [4,Theorem 4.12], since the proof of [4,Theorem 4.12], relying as it does on the DPRM, does not relativize to give an exactly parallel statement for n > 1.…”

“…Thus: A recurrent theme in studies of the relationships between classes of models of fragments (in one sense or another) of arithmetic has been cofinality of embeddings, it having been found that cofinality is often effective in either lifting properties from below or pulling them down from above. Theorem 4.1 of [4], the proof of which is rather routine, entails that any recursive ultrapower can be embedded properly, cofinally, and in A~ fashion in some other recursive ultrapower. In the A ~ case, the A~ of the embedding is automatic by virtue of the "DPRM" (or, as [1] prefers, "MRDP") theorem on diophantine representation of S ~ predicates; in the general case of A o ultrapowers, n__> 1, the A~ character of the embedding asserted in [4, Theorem 4.1] can be replaced by the stronger condition of n-elementariness, on account of a result of Dimitracopoulos and Gaifman [1 ] (Lemma 2.1 below).…”

“…For terminology and notation not already mentioned but to be used, the reader is for the most part referred to [4]; some will be explained as we go along.…”

“…If feff we shall denote by [f]~ the equivalence class of f modulo q/; the [,f]~ are the elements of ~-/0g.) These very special ultrapowers are models ofH ~ true arithmetic; indeed, they can be regarded (in a way pointed out in [-3] and [4]) as being the "fundamental" non-standard countable models of that theory.…”