The notion of slow provability for Peano Arithmetic (PA) was introduced by S.-D. Friedman, M. Rathjen, and A. Weiermann. They studied the slow consistency statement Con s asserting that a contradiction is not slow provable in PA. They showed that the logical strength of the theory PA + Con s lies strictly between that of PA, and PA together with its ordinary consistency: PA PA + Con s PA + Con PA .This paper is a further investigation into slow provability and its interplay with ordinary provability in PA. We study three variants of slow provability. The associated consistency statement of each of these yields a theory that lies strictly between PA and PA+Con PA in terms of logical strength. We investigate Turing-Feferman progressions based on these variants of slow provability. For our three notions, the Turing-Feferman progression reaches PA + Con PA in a different numbers of steps, namely ε 0 , ω, and 2. For each of the three slow provability predicates, we also determine its joint provability logic with ordinary PA-provability.

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This paper develops the philosophy and technology needed for adding a supremum operator to the interpretability logic ILM of Peano Arithmetic (PA). It is well-known that any theories extending PA have a supremum in the interpretability ordering. While provable in PA, this fact is not reflected in the theorems of the modal system ILM, due to limited expressive power. Our goal is to enrich the language of ILM by adding to it a new modality for the interpretability supremum. We explore different options for specifying the exact meaning of the new modality. Our final proposal involves a unary operator, the dual of which can be seen as a (nonstandard) provability predicate satisfying the axioms of the provability logic GL.