An introduction to recursively saturated and resplendent models

Abstract

The notions of recursively saturated and resplendent models grew out of the study of admissible sets with urelements and admissible fragments of Lω1ω, but, when applied to ordinary first order model theory, give us new tools for research and exposition. We will discuss their history in §3.The notion of saturated model has proven to be important in model theory. Its most important property for applications is that if , are saturated and of the same cardinality then = iff ≅ . See, e.g., Chang-Keisler [3]. The main drawback is that saturated models exist only under unusual assumptions of set theory. For example, if 2κ = κ+ then every theory T of L has a saturated model of power κ+. (Similarly, if κ is strongly inaccessible, then every T has a saturated model of power κ.) On the other hand, a theory T like Peano arithmetic, with types, cannot have a saturated model in any power κ with ω ≤ κ ≤ .One method for circumventing these problems of existence (or rather non-existence) is the use of “special” models (cf. [3]). If κ = Σλ<κ2λ, κ < ω, then every theory T of L has a special model of power κ. Such cardinals are large and, themselves, rather special. There are definite aesthetic objections to the use of these large, singular models to prove results about first order logic.

Axiomatizing first-order consequences in dependence logic

“…It is a direct consequence of the winning condition of player II that if X is the set of assignments s with s(y i ) = f i (s(x 1 ), ..., s(x m )), 1 ≤ i ≤ n, then (20) holds. There are many recursively saturated models: Proposition 14 ( [2]). For every infinite A there is a recursively saturated count-…”

On maximum conservative extensions

“…The same question could be asked of other natural families of fragments of arithmetic, so we also wonder what the limits to this result are. Over ACA 0 there is a known theory of Π 1 3 , Π 1 2 conservative extensions (see [13]), some of it developed using techniques which inspired ideas in this paper (especially [2,3]). Analogously to Π 1 1 -MAX, there are largest Π 1 n+1 , Π 1 n conservative theories for all n ≥ 1, and these theories are, by the same argument as for n = 1, Π 2 .…”

The proof-theoretic strength of Ramsey's theorem for pairs and two colors

“…To show this theorem, we use the following property of recursively saturated models and resplendent models, which are introduced by Barwise and Schlipf. See [4] for the historical information of recursively saturated models and resplendent models. Theorem 6.2 (see Sections 1.8 and 1.9 of [38]).…”