DESCRIPTIVE INNER MODEL THEORY

Project Details

Description

The main open problem of inner model theory is the construction of canonical inner models with large cardinals with supercompact cardinals being the main target. In 60s and 70s, major progress was made towards the resolution of the problem. In particular, such canonical models were constructed for many large cardinals in the region of measurable cardinals and strong cardinals. However, soon new obstacles were discovered and through a fundamental work done by Martin, Steel and Woodin, it became increasingly clear that the ultimate resolution of the problem has to incorporate ideas from descriptive set theory. In late 80s and early 90s, Martin, Steel and Woodin, discovered many bridges between inner model theory and descriptive set theory thus unifying the two areas of set theory. Descriptive inner model theory is the theory that has emerged through their work. Its main technical problem is the descriptive version of the inner model problem, namely, the problem of constructing canonical models capturing the truth. While many formal versions of this problem are available, the one that has been publicized most is the Mouse Set Conjecture (MSC) which conjectures that in models of determinacy, ordinal definability, the most powerful form of definability, can be captured via canonical models of set theory. Sargsyan, in his thesis, building on an earlier work of Woodin, developed techniques for proving MSC and used these techniques to obtained some partial results on MSC. These partial results then were translated into the ordinary language of inner model theory, producing canonical models with large cardinals. Sargsyan believes that the descriptive approach to the inner model problem is the most promising route to the resolution of the 50 year old inner model problem and he plans to pursue it further.Set theory is the language of mathematics. It provides the basic foundations above which the rest of mathematics is build. One of the most basic problems in the foundations of mathematic has been the reduction of all of mathematics into basic set of axioms whose consistency can either be proven or convincingly argued for. Gödel's celebrated incompleteness theorems imply that the first option is impossible. However, because of major developments in the previous century, the second option is possible via a class of axioms of infinity known as large cardinal axioms which assert the existence of very large sets. Virtually, as Gödel himself predicted, every know natural mathematical theory has been reduced to some large cardinal axiom. The set theorists of the 21st century inherited the problem of arguing for the consistency of such axioms which is usually done by exhibiting canonical examples or rather models of such axioms. The problem of exhibiting such canonical models for large cardinals has been known as the inner model problem which is the main subject of the proposed research.
StatusFinished
Effective start/end date6/1/125/31/15

Funding

  • National Science Foundation (National Science Foundation (NSF))

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