Descriptive Inner Model Theory

Rutgers University New Brunswick, New Brunswick NJ

Investigators

Abstract

Set theory is a relatively new branch of mathematics that has its roots in mathematical analysis and philosophy. Its philosophical beginning had to do with the foundations of mathematics, which at the end of the 19th century and at the beginning of the 20th century was not standing on firm grounds. Set theorists of the time devised an axiomatic system known as Zermelo-Fraenkel with the Axiom of Choice (ZFC), that removed all paradoxes of the time, and put all of mathematics on firmer grounds. On the other hand, Cantor, working on questions coming from the study of Fourier sequences, needed to device longer inductions then the set of natural numbers allowed him to do. His work led him to what is now known as cardinal numbers, ordinal numbers and above all the study of the real numbers. His work led to what is called The Continuum Hypothesis, which was the first open problem on Hilbert's infamous list of problems. Godel and Cohen eventually showed that The Continuum Hypothesis is undecidable, i.e. neither provable nor disprovable, from, ZFC, thus exposing a major weakness in the axiomatic system ZFC. It does not decide simple questions about sets of reals, one of the most basic mathematical objects. Descriptive Inner Model Theory is one part of what is now called Godel's Program, which is the program of removing undecidability from mathematics by studying stronger and stronger natural axiomatic systems extending ZFC that decide more and more natural questions about sets of reals and other mathematical objects. Godel himself suggested that the Large Cardinal Hierarchy should be used for the aforementioned purpose. The goal of descriptive inner model theory is to build canonical models for large cardinals and also study their impact on the set of real numbers. For example, one of the greatest achievements of the subject due to many authors is that given large cardinals, projective sets of reals behave exactly as Borel sets of reals as far as their regularity properties are concerned (i.e. measurability, category and etc). In addition the project also provides research training opportunities for graduate students. The PI is proposing to attack some of the central open problems of the area such as the HOD Problem or the Iterability Problem. The HOD Problem essentially says that in models where sets of reals posses all regularity properties one may desire, the universe consisting of sets that are hereditarily definable from ordinals is a canonical inner model for large cardinals. It connects the desirable regularity properties of sets of reals with the existence of large cardinals in a fundamental way showing that the regularity properties of definable sets of reals are a consequence of large cardinals and vice versa. The HOD Problem was isolated in late 90s and early 2000s as one of the most central open problems of the area. Iterability Problem is of similar nature but avoids determinacy models. This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.