Analysis of Martin-Harrington Theorem in Higher Order Arithmetic

Abstract

The main effort in this thesis is to answer some questions from Professor W.Hugh Woodin about Martin-Harrington theorem. All proofs of Harrington's theorem $``Det(\Sigma_1^1)$ implies $0^{\sharp}$ exists" we know are proved in two steps: firstly show that $``Det(\Sigma_1^1)$ implies \textbf{Harrington's $\star$}" and secondly derive the existence of $0^{\sharp}$ from \textbf{Harrington's $\star$} by Silver's theorem. We observe that $``Z_2+Det(\Sigma_1^1)$ implies \textbf{Harrington's $\star$}". A natural question is ``whether $Z_2+$\textbf{Harrington's $\star$} implies $0^{\sharp}$ exists". We show that $Z_2+$ \textbf{Harrington's $\star$} does not imply $0^{\sharp}$ exists. Furthermore, we show that $Z_3+$ \textbf{Harrington's $\star$} does not imply $0^{\sharp}$ exists. As a corollary of $``Z_4+$ \textbf{Harrington's $\star$} implies $0^{\sharp}$ exists", $Z_4$ is the minimal system in higher order arithmetic to prove ``\textbf{Harrington's $\star$} implies $0^{\sharp}$ exists". As a corollary of $``Z_4+$ \textbf{Harrington's $\star$} implies $0^{\sharp}$ exists", lightface Harrington's theorem is provable in $Z_4$. We show that boldface Harrington's theorem is provable in $Z_2$.