Please use this identifier to cite or link to this item: http://scholarbank.nus.edu.sg/handle/10635/36557
Title: Analysis of Martin-Harrington Theorem in Higher Order Arithmetic
Authors: CHENG YONG
Keywords: Harrington's theorem, \textbf{Harrington's $\star$}, $0^{\sharp}$, almost disjoint forcing, Baumgartner's forcing, strong reflecting property
Issue Date: 10-Jul-2012
Source: CHENG YONG (2012-07-10). Analysis of Martin-Harrington Theorem in Higher Order Arithmetic. ScholarBank@NUS Repository.
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 $Z_2$.
URI: http://scholarbank.nus.edu.sg/handle/10635/36557
Appears in Collections:Ph.D Theses (Open)

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