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Project number   Stand-alone Projects  P16334
Title   Pure Set Theory: Inner Models, Forcing and Absoluteness
Principal investigator   FRIEDMAN Sy David
Approval date   03.03.2003
University / Research institution   Kurt Gödel Research Center for Mathematical Logic, Universität Wien
Scientific field(s)  
Keywords   Constructibility, Forcing, Extenders, Large Cardinals, Projective Sets, Absoluteness
Homepage   http://www.logic.univie.ac.at/Home.html


Mathematical logic entered the modern era through the work of Kurt Gödel. This project is part of the ongoing revival of the Gödel tradition at the University of Vienna, where Gödel established his famous Completeness and Incompleteness theorems in the 1930's. Our topic is set theory, the area of logic that most interested Gödel in his mature years.
Set theory today exhibits two interconnected aspects, the pure and the applied. The former is concerned with the analysis of infinity, leading to a picture of the universe of sets as a whole, whereas the latter refers to the many successes of pure set theory either in solving mathematical problems or in showing that they are unsolvable using the traditional axioms of set theory. This project is concerned with pure set theory, and will explore the following topics: constructibility, iterated forcing, class forcing, inner model theory and absoluteness principles.
In constructibility, we will discuss some new combinatorial principles that hold in Gödel's model and further develop the hyperfine structure theory. In iterated forcing, we will consider iterations indexed by morasses and develop a generalization of properness that applies not only to set-forcing. In class forcing we will look for a valid version of Solovay's dichotomy, a forcing which provably has a unique generic and further investigate generic saturation. In inner model theory, we will consider new extender model constructions, the new coding method in the presence of Woodin cardinals and connections with the theory of projective sets. And in the theory of absoluteness we will study the bounded Martin Maximum and develop an absoluteness principle appropriate for the theory of class forcing.



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