Determinacy and Fast-growing Sequences of Turing Degrees

Abstract

We discuss sufficiently fast-growing sequences of Turing degrees. The key result is that, assuming sufficient determinacy, if $\varphi$ is a formula with one free variable, and S and T are sufficiently fast-growing sequences of Turing degrees of length ${\omega }_1$, then $\varphi (S)⟺\varphi (T)$. 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.

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