Finitely Axiomatized Theories Lack Self-Comprehension

TL;DR

This paper demonstrates that any consistent finitely axiomatized one-dimensional theory cannot fully interpret its own extension with predicative comprehension, extending incompleteness results without relying on arithmetic.

Contribution

It proves a new form of incompleteness applicable to arbitrary weak theories, independent of base theories, using novel forcing-interpretation methods.

Findings

No finitely axiomatized one-dimensional theory interprets its predicative extension

The result applies broadly to weak theories, not just extensions of a base theory

Introduces a new perspective on sequential theories via forcing-interpretations

Abstract

In this paper we prove that no consistent finitely axiomatized theory one-dimensionally interprets its own extension with predicative comprehension. This constitutes a result with the flavor of the Second Incompleteness Theorem whose formulation is completely arithmetic-free. Probably the most important novel feature that distinguishes our result from the previous results of this kind is that it is applicable to arbitrary weak theories, rather than to extensions of some base theory. The methods used in the proof of the main result yield a new perspective on the notion of sequential theory, in the setting of forcing-interpretations.