An intensional fixed point theory over first order arithmetic

The purpose of this article is to present a new theory IPA(σ) for fixed points over arithmetic which allows the building up of fixed points in a very nested and entangled way. But in spite of its great expressive power we can show that the proof-theoretic strength of our theory—which is intensional in a meaning to be described below—is characterized by the Feferman–Schütte ordinal Γ0. Our approach is similar to the building up of fixed points over state spaces in the propositional modal μ-calculus.