On a question of Krajewski's

TL;DR

This paper demonstrates the existence of conservative extensions with specific interpretability properties in expanded languages of recursively enumerable theories, providing a negative answer to a question posed by Krajewski.

Contribution

It establishes that for certain expansions, conservative extensions can be ordered by interpretability in a way that contradicts previous assumptions, showing limitations in the structure of such theories.

Findings

Existence of conservative extensions with interpretability properties

Failure of the result with unary predicate expansions

Preservation of results under model-conservative extensions

Abstract

In this paper we provide a (negative) solution to a problem posed by Stanis{\l}aw Krajewski. Consider a recursively enumerable theory U and a finite expansion of the signature of U that contains at least one predicate symbol of arity $\ge$ 2. We show that, for any finite extension $\alpha$ of U in the expanded language that is conservative over U, there is a conservative extension $\beta$ of U in the expanded language, such that $\alpha\vdash\beta$ and $\beta\nvdash\alpha$. The result is preserved when we consider either extensions or model-conservative extensions of U in stead of conservative extensions. Moreover, the result is preserved when we replace $\vdash$ as ordering on the finitely axiomatized extensions in the expanded language by a special kind of interpretability, to wit interpretability that identically translates the symbols of the U-language. We show that the result fails when we consider an expansion with only unary predicate symbols for conservative extensions of U ordered by interpretability that preserves the symbols of U.