Can we make the second incompleteness theorem coordinate free?

Abstract

Is it possible to give a coordinate free formulation of the Second Incompleteness Theorem? We pursue one possible approach to this question. We show that (i) cutfree consistency for finitely axiomatized theories can be uniquely characterized modulo EA-provable equivalence, (ii) consistency for finitely axiomatized sequential theories can be uniquely characterized modulo EA-provable equivalence. The case of infinitely axiomatized ce theories is more delicate. We carefully discuss this in the paper.