On Shavrukov's non-isomorphism theorem for diagonalizable algebras

Abstract

We prove a strengthened version of V. Yu. Shavrukov's result on the non-isomorphism of diagonalizable algebras of two $\Sigma_1$-sound theories, based on the improvements previously found 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 is epimorphism from $(\mathfrak{L}_T, \Box_T \Box_T)$ onto $(\mathfrak{L}_T, \Box_T)$. The case of bimodal diagonalizable algebras is also considered. We give several examples of pairs of theories with isomorphic diagonalizable algebras but non-isomorphic bimodal diagonalizable algebras.