Diagonalizable algebras and the length of proofs

Abstract

We rederive a theorem of Shavrukov [17] on the diagonaliz- able algebras of PA and ZF using an approach that differs ever-so slightly from the original. This gives a somewhat stronger result (which was im- plicit in the original proof) that we, coupled with a more careful analysis of the Parikh speed-up phenomenon, put to use by giving examples of how different choices of provability predicates for a given theory T can lead to non-isomorphic algebras. We also show that, by a minor tweak, the in- jectivity assumption can be dropped and the result extended to arbitrary epimorphisms.