The PI is investigating the theory of large cardinals, and their applications to determinacy, with emphasis on the following topics: (1) long games and iterability; (2) forcing with ultrafilters over models of determinacy; and (3) monadic theories of ordinals. 1 -- Previous work by the PI identified a specific class of games of uncountable length, so that the associated game quantifier is precisely strong enough to define the minimal iterable inner model with an external measure concentrating on Woodin cardinals. This is the least level in the large cardinal hierarchy which cannot be captured by games of countable length. The present project aims to extend and study this connection between levels of the large cardinal hierarchy and games of uncountable length. 2 -- Using inner models for large cardinals it is possible to identify ultrafilters on specific sets of countable sequences in the smallest model of set theory containing all the reals. These ultrafilters give rise to interesting forcing extensions of the model. The PI is investigating the constructions of ultrafilters from large cardinals with the aim of generalizing them to the case of uncountable sequences, and studying the resulting forcing extensions. 3 --- The PI is studying the expressive power of the monadic second order language in the structure of the ordinals, both under the axiom of choice (for ordinals above the second uncountable cardinal) and under the axiom of determinacy.

Large cardinal axioms state the existence of functions which act on the entire universe of sets, and preserve the structure of set membership. It is one of the most amazing discoveries of modern set theory that these functions, which at face value should only affect extremely large sets (large enough to not be definable from smaller sets using the structure of set membership), concretely affect the properties of real numbers. The intermediary connecting large cardinals to real numbers is the axiom of determinacy, stating the existence of winning strategies in infinite games of perfect information. The present project is part of the study of the ties between large cardinals and determinacy. It addresses games of uncountable infinite length, two-valued measures on sequences of uncountable length under the axiom of determinacy, and the expressive power of statements involving sets, but not functions, over wellordered structures.

Project Report

This project supported research by the PI and doctoral students in mathematics. The research conducted is in mathematical logic, and within mathematical logic it is mostly in set theory. It seeks to determine possible behaviors of the universe of mathematics, and the relative strength of mathematical assertions. Our knowledge of the universe of mathematics comes through deduction from a system of axioms called ZFC. By Goedel's incompleteness theorem we can never expect to obtain complete knowledge of the universe of mathematics. There are always assertions which are independent of the axioms, meaning that they can neither be proved, nor refuted. The independence of specific assertions can be shown by constructing models for the axioms where the assertions hold, and other models where they fail. A central technique for doing this was developed by Cohen, who termed it forcing. It allows expanding universes which satisfy ZFC. Much of the research done in this project used forcing to obtain independence results in set theory. Among the main results is a proof that establishes the independence of minimality conditions on the powerset function from an important property in infinitary combinatorics, the tree property. Another result relates two hierarchies of axioms, one that involves reflection properties of the universe of mathematics, and another that involves saturation under varying classes of forcing. Other results apply forcing outside set theory, to the field of reverse mathematics. This field seeks to determine the strength of assertions of mathematics that involve sets of natural numbers. Strength is measured on a hierarchy of logical axioms about set existence. Research under this grant used forcing to establish the strength of a particular assertion, that provides several ``firsts'' in the field. In particular it is the first example where it is known that the strength cannot be measured accurately without strong induction assumptions. Additional results develop the technique of forcing itself, providing new ways to iterate forcing constructions. Iterations are useful because they allow breaking a construction into small, more easily manageable parts. The new iteration techniques developed under this proposal should make it possible to explore a broad array of independence results that have been beyond reach before.

Agency
National Science Foundation (NSF)
Institute
Division of Mathematical Sciences (DMS)
Application #
0556223
Program Officer
Tomek Bartoszynski
Project Start
Project End
Budget Start
2006-04-15
Budget End
2011-06-30
Support Year
Fiscal Year
2005
Total Cost
$377,236
Indirect Cost
Name
University of California Los Angeles
Department
Type
DUNS #
City
Los Angeles
State
CA
Country
United States
Zip Code
90095