Phase transitions in axiomatic thought [Elektronische Ressource] / vorgelegt von Gyesik Lee

Gyesik LeePhase Transitions in Axiomatic Thought2005MathematikPhase Transitions in Axiomatic ThoughtInaugural-Dissertationzur Erlangung des Doktorgradesder Naturwissenschaften im FachbereichMathematik und Informatikder Mathematisch-Naturwissenschaftlichen Fakult¨atder Westf¨alischen Wilhelms-Universit¨at Mu¨nstervorgelegt vonGyesik Leeaus Kyunggi-Do, Su¨dkorea– 2005 –Dekan: Prof. Dr. K. HinrichsErster Gutachter: Prof. Dr. A. WeiermannZweiter Gutachter: Prof. Dr. R. D. SchindlerTag der mu¨ndlichen Pru¨fung: 29. 04. 2005Tag der Promotion: 13. 07. 2005iiTo my familyandall my Ourtown friendsAbstractAn aspect of the thesis is to investigate well-known ordinal notation systems forPA. It will be shown that the so-called phase transition phenomenon can beobserved, i.e., there are thresholds between provability and unprovability. Thisinvestigation leads to a comparison of the ordinal notation systems.The thesis gives also a guide how one can generally establish such phase tran-sitions in every logic system which is strong enough in the sense of G¨odel. Weshall see that Friedman style miniaturizations play the central role.Another point of the thesis is the parametrized version of the Kanamori-McAloon principle. This variants of the finite Ramsey theorem is equivalent totheParis-Harringtonprinciple.