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 Citation: 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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