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ON A QUESTION OF KRAJEWSKI’S

Published online by Cambridge University Press:  14 March 2019

FEDOR PAKHOMOV  and
ALBERT VISSER
FEDOR PAKHOMOV
Affiliation:
DEPARTMENT OF MATHEMATICAL LOGIC STEKLOV MATHEMATICAL INSTITUTE 8 GUBKINA STREET, MOSCOW 119991RUSSIAE-mail: pakhfn@mi.ras.ru
ALBERT VISSER
Affiliation:
PHILOSOPHY, FACULTY OF HUMANITIES UTRECHT UNIVERSITY, JANSKERKHOF 13 3512BL UTRECHT, THE NETHERLANDSE-mail: a.visser@uu.nl
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Abstract

In this paper, we study finitely axiomatizable conservative extensions of a theory U in the case where U is recursively enumerable and not finitely axiomatizable. Stanisław Krajewski posed the question whether there are minimal conservative extensions of this sort. We answer this question negatively.

Consider a finite expansion of the signature of U that contains at least one predicate symbol of arity ≥ 2. We show that, for any finite extension α of U in the expanded language that is conservative over U, there is a conservative extension β of U in the expanded language, such that $\alpha \vdash \beta$ and $\beta \not \vdash \alpha$. The result is preserved when we consider either extensions or model-conservative extensions of U instead of conservative extensions. Moreover, the result is preserved when we replace $\dashv$ as ordering on the finitely axiomatized extensions in the expanded language by a relevant 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.

Keywords

03B1003B3003F25predicate logicrecursively enumerable theoriesinterpretabilitylinear orderings
Type
Articles
Information
The Journal of Symbolic Logic , Volume 84 , Issue 1 , March 2019 , pp. 343 - 358
Copyright
Copyright © The Association for Symbolic Logic 2019 

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