The Inner Model Program is one of the central areas of Set Theory. The traditional view of this program has been that of an increment program with progress measured by a progression up the hierarchy of large cardinals. It is now known that this view is not correct. A critical transition occurs at the level of exactly one supercompact cardinal, and in solving the Inner Model Problem for this specific large cardinal axiom, one solves the Inner Model Problem for essentially all known large cardinal axioms. The solution must necessarily yield an ultimate enlargement of Godel's inner model L. This Ultimate-L must closely approximate the parent universe V within which it is constructed. Though the detailed construction of Ultimate-L is open, the axiom, ``V = Ultimate-L'', can be precisely formulated and its consequences explored even now. One can also precisely specify a conjecture which would be the final outcome of the construction of Ultimate-L. This is the Ultimate-L Conjecture. This conjecture is closely related to several other conjectures and these collectively are the focus of this research proposal.

The mathematical study of Infinity is the focus of Set Theory. This subject began on the basis of principles isolated by Cantor and then elaborated on by others. By the early part of the 20th century the basic Zermelo-Frankel Axioms had been isolated and these axioms together with the Axiom of Choice are the ZFC axioms. The ZFC axioms define the current conception of (mathematical) Infinity. The seminal discoveries of the latter half of the 20th century showed that most of the fundamental questions asked about infinite sets are unsolvable on the basis of the ZFC axioms. Famous among these unsolvable questions is that of Cantor's Continuum Hypothesis which by the middle of the 20th century was widely regarded as one of the most important questions of mathematics. This proposal focuses on the Ultimate-L Project. This project if successful will isolate (for the first time) an extension of the ZFC axioms that will resolve essentially all of these otherwise unsolvable questions (including that of the Continuum Hypothesis) and provide a framework for reducing all questions of Set Theory to questions of the existence of large infinite sets--these are the so called Axioms of Strong Infinity. The new approach arises from a series of recent results which show that such an extension of the ZFC axioms might actually exist.

National Science Foundation (NSF)
Division of Mathematical Sciences (DMS)
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Tomek Bartoszynski
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Harvard University
United States
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