Nonstandard characterizations of recursive saturation and resplendency

We prove results about nonstandard formulas in models of Peano arithmetic which complement those of Kotlarski, Krajewski, and Lachlan in [KKL] and [L]. This enables us to characterize both recursive saturation and resplendency in terms of statements about nonstandard sentences. Specifically, a model of PA is recursively saturated iff is nonstandard and -logic is consistent. is resplendent iff is nonstandard, -logic is consistent, and every sentence φ which is consistent in -logic is contained in a full satisfaction class for . Thus, for models of PA, recursive saturation can be expressed by a (standard) -sentence and resplendency by a -sentence.

Blunt and topless end extensions of models of set theory

Let be a well-founded model of ZFC whose class of ordinals has uncountable cofinality, such that has a Σn end extension for each n ∈ ω. It is shown in Theorem 1.1 that there is such a model which has no elementary end extension. In the process some interesting facts about topless end extensions (those with no least new ordinal) are uncovered, for example Theorem 2.1: If is a well-founded model of ZFC, such that has uncountable cofinality and has a topless Σ3 end extension, then has a topless elementary end extension and also a well-founded elementary end extension, and contains ordinals which are (in ) highly hyperinaccessible. In §3 related results are proved for κ-like models (κ any regular cardinal) which need not be well founded. As an application a soft proof is given of a theorem of Schmerl on the model-theoretic relation κ → λ. (The author has been informed that Silver had earlier, independently, found a similar unpublished proof of that theorem.) Also, a simpler proof is given of (a generalization of) a characterization by Keisler and Silver of the class of well-founded models which have a Σn end extension for each n ∈ ω. The case κ = ω1 is investigated more deeply in §4, where the problem solved by Theorem 1.1 is considered for non-well-founded models. In Theorems 4.1 and 4.4, ω1-like models of ZFC are constructed which have a Σn end extension for all n ∈ ω but have no elementary end extension. ω1-like models of ZFC which have no Σ3 end extension are produced in Theorem 4.2. The proof uses a notion of satisfaction class, which is also applied in the proof of Theorem 4.6: No model of ZFC has a definable end extension which satisfies ZFC. Finally, Theorem 5.1 generalizes results of Keisler and Morley, and Hutchinson, by asserting that every model of ZFC of countable cofinality has a topless elementary end extension. This contrasts with the rest of the paper, which shows that for well-founded models of uncountable cofinality and for κ-like models with κ regular, topless end extensions are much rarer than blunt end extensions.

Construction of Satisfaction Classes for Nonstandard Models

Given a resplendent model for Peano arithmetic there exists a full satisfaction class over , i.e. an assignment of truth-values, to all closed formulas in the sense of with parameters from , which satisfies the usual semantic rules. The construction is based on the consistency of an appropriate system of -logic which is proved by an analysis of standard approximations of nonstandard formulas.

Full Satisfaction Classes and Recursive Saturation

It is shown that a nonstandard model of Peano arithmetic which has a full satisfaction class is necessarily recursively saturated.