Model Theory of the Inaccessibility Scheme

TL;DR

This paper explores the model-theoretic properties of theories satisfying the Inaccessibility Scheme, focusing on elementary end extensions in models with inaccessible cardinals, advancing understanding of large cardinal implications in model theory.

Contribution

It introduces a detailed study of the model theory related to the Inaccessibility Scheme, particularly regarding elementary end extensions in models with inaccessible cardinals.

Findings

Characterization of models with inaccessible cardinals

Existence results for elementary end extensions

Insights into the structure of theories satisfying the Inaccessibility Scheme

Abstract

Suppose L = {<, . . .} is any countable first order language in which < is interpreted as a linear order. Let T be any complete first order theory in the language L such that T has a kappa-like model where kappa is an inaccessible cardinal. Such T satisfies the Inaccessibility Scheme. In this paper we study model theory of the inaccessibility scheme at the level of the existence of elementary end extensions for various models of it.