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ExoticPrefix091406

ExoticPrefix091406 - 1 EXOTIC PREFIX THEORY abstract by...

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1 EXOTIC PREFIX THEORY abstract by Harvey M. Friedman [email protected] September 14, 2006 (This corrects an earlier version with the same date that has been removed from my website). The state of the art on Order Calculus and the variants of the P 01 independent statements that are associated with the subtle cardinal hierarchy (not the Mahlo cardinal hierarchy), is now contained in Search for Consequences, under downloadable manuscripts, Lecture Notes, http://www.math.ohio-state.edu/%7Efriedman/ Note that the independent statements in Search for Consequences go forward and backward. The ones in posting #299 only go forward, and we are no longer confident that we can get away with this in #299. ############################ The Order Calculus development now appears as key Lemmas in Exotic Prefix Theory. Exotic Prefix Theory is the study of basic prefix classes, comprising finitely many sentences, at least one of which is independent of ZFC. Such a prefix class is said to be "exotic". The goal is to show that various exotic prefix classes can be "tamed by large cardinals". I.e., every statement in the class is either provable or refutable using presently formulated large cardinals. Some of these exotic prefix classes consist entirely of explicitly P 01 sentences. At the moment, this goal is out of reach for any basic exotic prefix class. However, we expect a long series of partial results, that various subclasses can be tamed by various large cardinals.
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2 In both Search for Consequences, and Forty Years on his Shoulders, under downloadable manuscripts, http://www.math.ohio- state.edu/%7Efriedman/ we made the following conjecture: EXOTIC CONJECTURE Every interesting mathematical theorem can be recast as one among a natural finite set of statements, all of which can be decided using well studied extensions of ZFC, but not within ZFC itself. We now present some prefix classes that comprise finitely many statements, at least one of which is independent of ZFC (exotic prefix classes). Some of the prefix classes consist entirely of explicitly P 01 sentences.
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