Incompleteness of boundedly axiomatizable theories

TL;DR

This paper proves that any consistent, finitely axiomatized sequential theory with bounded quantifier complexity is incomplete, using a reduction approach and Tarski's Undefinability of Truth theorem.

Contribution

It establishes a new incompleteness result for theories with bounded quantifier alternation, extending classical incompleteness to a broader class of theories.

Findings

Any such theory is incomplete.

The proof employs a reduction mechanism and Tarski's theorem.

Additional incompleteness results are derived.

Abstract

Our main result (Theorem A) shows the incompleteness of any consistent sequential theory T formulated in a finite language such that T is axiomatized by a collection of sentences of bounded quantifier-alternation-depth. Our proof employs an appropriate reduction mechanism to rule out the possibility of completeness by simply invoking Tarski's Undefinability of Truth theorem. We also use the proof strategy of Theorem A to obtain other incompleteness results (as in Theorems A+; B and B+).