Incompleteness and jump hierarchies.

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) ∈ R² : O^A &#8804:_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, if the rank of A is α, then ω₁^A is the (1 + α)th admissible ordinal. It follows, assuming suitable large cardinal hypotheses, that, on a cone, the rank of X is ω₁^X. [ABSTRACT FROM AUTHOR]