Incompleteness of boundedly axiomatizable theories
arXiv:2311.14025 · doi:10.1090/proc/16975
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+).
arXiv comment6 pages; in this version reference to MathOverflow work of Emil Jeřábek has been added, and the author list of [AGLRZ] is now up-to-date