Decidability in the Hyperdegrees and a Theorem of Hyperarithmetic Analysis

Abstract

In this thesis we explore two different topics: the complexity of the theory of the hyperdegrees, and the reverse mathematics of a result in graph theory. For the first, we show the $\Sigma_{2}$ theory of the hyperdegrees as an upper-semilattice is decidable, as is the $\Sigma_{2}$ theory of the hyperdegrees below Kleene's $\mathcal{O}$ as an upper-semilattice with greatest element. These results are related to questions of extensions of embeddings into both structures, i.e., when do embeddings of a structure extend to embeddings of a superstructure. The second part is joint work with Richard Shore and Jun Le Goh. We investigate a theorem of graph theory and find that one formalization is a theorem of hyperarithmetic analysis: the second such example found, as it were, in the wild. This work is ongoing, and more may appear in future publications.