INNER MODEL THEORY · Inner Models of Set Theory, large cardinals and fine structure theory

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Project objective

The proposed project is devoted to inner model theory, an area of set theory, and its applications in infinitary combinatorics and descriptive set theory. The main goal of inner model theory is constructing canonical models, or extender models, for axioms postulating the existence of 'higher level infinity', the so-called large cardinal axioms.Extender models are used (a) to establish consistency strengths for mathematical statements that cannot be decided by means of the standard axioms of set theory, the Zermelo-Fraenkel axioms, and (b) to extract information about sets that depends on large cardinals alone.The project is divided into 4 parts:The 1. part focuses on the analysis of the internal structure of extender models and studying combinatorial principles in these models using purely fine structural methods. The goal here is to generalize Jensen and analysis of Goedel's constructible universe to the broader context of extender models.The 2. part is devoted to applications of the existing inner model theory in improving the known lower bounds for consistency strengths of various set-theoretic statements, such as the failure of Jensen and principle Diamond at Mahlo cardinals under GCH, failure of Jensen and principle square at singular cardinals, or bounded Martin Maximum.The 3. part concentrates on connections between inner model theory and descriptive set theory; the objectives here include- establishing optimal hypotheses for correctness results at higher level of projective hierarchy and- finding inner model characterizations of objects known from classical descriptive set theory.The 4. part is devoted to constructions of inner models, and the emphasis is on studying the complexities of iteration strategies for countable extender models and determining limitations of currently known methods for obtaining iterability.

Original text from CORDIS.

Participants

• WISSENSCHAFTSKOLLEG ZU BERLIN - INSTITUTE FOR ADVANCED STUDY · BERLINCoordinatorCity levelGermany

Links

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