Incompleteness and jump hierarchies

Abstract

This paper is an investigation of the relationship between Gödel’s second incompleteness theorem and the well-foundedness of jump hierarchies. It follows from a classic theorem of Spector that the relation { ( A , B ) ∈ R 2 : O A ≤ H B } \{(A,B) \in \mathbb {R}^2 : \mathcal {O}^A \leq _H B\} is well-founded. We provide an alternative proof of this fact that uses Gödel’s second incompleteness theorem instead of the theory of admissible ordinals. We then derive a semantic version of the second incompleteness theorem, originally due to Mummert and Simpson, from this result. Finally, we turn to the calculation of the ranks of reals in this well-founded relation. We prove that, for any A ∈ R A\in \mathbb {R} , if the rank of A A is α \alpha , then ω 1 A \omega _1^A is the ( 1 + α ) (1 + \alpha ) th admissible ordinal. It follows, assuming suitable large cardinal hypotheses, that, on a cone, the rank of X X is ω 1 X \omega _1^X .