This award is funded under the American Recovery and Reinvestment Act of 2009 (Public Law 111-5).

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 Program for this specific large cardinal axiom, one solves the Inner Model Program for essentially all known large cardinal axioms. The solution must necessarily yield an ultimate enlargement of Goedel's inner model L. This ultimate-L must closely approximate the parent universe within which it is constructed. The ramifications for Set Theory will be profound, ranging from new inconsistency results in ZF to combinatorial theorems of ZFC proved by exploiting the closeness of ultimate-L to V. Ultimate-L will also provide a key setting for resolving the hierarchy of large cardinals beyond the level of omega-huge cardinals which constitute essentially the strongest large cardinals axioms which are not known to be inconsistent. The point is that ultimate-L inherits large cardinals from V exactly as L inherits large cardinals from V if 0-sharp does not exist. Therefore the analysis of what is possible in ultimate-L is in effect the analysis of what is possible in V.

The mathematical study of Infinity in its modern incarnation dates from the work of Cantor in the late 19th century, this area of mathematics is Set Theory. With the results of Goedel and then Cohen from the middle of the 20th century it was established that many of the fundamental problems could not be solved on the basis of the current ZFC axioms for Set Theory. The most famous example is the problem of Cantor's Continuum Hypothesis which was placed first by Hilbert on his list of 20 questions in 1900. The solution to this question and the numerous other questions now known to be unsolvable requires the discovery of new axioms beyond the current axioms. Over the last few years a new approach to this problem has emerged based on generalizations of Goedel's axiom of constructibility. There is now convincing evidence that this approach will lead to new axioms which settle all the problems of Set Theory currently known to be unsolvable, are compatible with all known strong axioms of infinity, and which themselves are immune to kind of unsolvable problems that are ubiquitous in Set Theory. These examples will be the first examples of such axioms ever discovered and therefore show great promise for at long last finding the correct axioms for Set Theory and thereby solving in particular the problem of the Continuum Hypothesis.

National Science Foundation (NSF)
Division of Mathematical Sciences (DMS)
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Tomek Bartoszynski
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University of California Berkeley
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