On a Strengthening of the Non-Isomorphism Theorem for Provability Algebras

We prove a strengthened version of V. Shavrukov’s result on the non-isomorphism of provability algebras of two Σ1-sound theories, based on the improvements previously discovered by G. Adamsson. We then obtain several corollaries to the strengthened result by applying it to various pairs of theories and obtain new non-isomorphism examples. In particular, we show that there are no epimorphisms from $$({{\mathfrak{L}}_{T}},\,{{\square }_{T}}{{\square }_{T}})$$ onto $$({{\mathfrak{L}}_{T}},{{\square }_{T}})$$ .