Incompleteness of boundedly axiomatizable theories
Ali Enayat, Albert Visser · Proceedings of the American Mathematical Society · 2024
Our main result (Theorem A) shows the incompleteness of any consistent sequential theory T T formulated in a finite language such that T 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.