Isomorphisms of diagonalizable algebras

For a formal theory T the diagonalizable algebra a k a Magari algebra of T denoted DT is the Lindenbaum sentence algebra of T endowed with the unary operator T arising from the provability predicate of T the equivalence class of a sentence is sent by T to the equivalence class of the T sentence expressing that T proves It was shown in Shavrukov that the diagonalizable algebras of PA and ZF as well as the diagonalizable algebras of similarly related pairs of sound theories are not isomorphic Neither are these algebras rst order equivalent Shavrukov Theorem In the present paper we establish a su cient condition which we name B co herence for the diagonalizable algebras of two theories to be isomorphic It is then immediately seen that DZF DGB which answers a question of Smory nski We also construct non identity automorphisms of diagonalizable algebras of all theories un der consideration The techniques we use are a combination of those developed in the context of partially conservative sentences cf Lindstr om and those of Pour El Kripke A related construction appears in Solovay Theorem