Determinacy and Fast-growing Sequences of Turing Degrees

TL;DR

This paper explores the properties of fast-growing sequences of Turing degrees under determinacy assumptions, showing they are indistinguishable for certain formulas and degrees, and extends these ideas to degrees of subsets of .

Contribution

It establishes that under sufficient determinacy, sequences of Turing degrees of length  are equivalent with respect to formulas, and introduces degrees for subsets of .

Findings

Sequences of Turing degrees of length  are indistinguishable under determinacy.

High degrees of subsets of  are effectively indistinguishable under determinacy and CH.

Abstract

We discuss sufficiently fast-growing sequences of Turing degrees. The key result is that, assuming sufficient determinacy, if $\phi$ is a formula with one free variable, and S and T are sufficiently fast-growing sequences of Turing degrees of length $\omega_1$, then $\phi(S) \iff \phi(T)$. We also define degrees for subsets of $\omega_1$ analogous to Turing degrees, and prove that under sufficient determinacy and CH, all sufficiently high degrees are also effectively indistinguishable.