Project number 

Standalone 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 setforcing. 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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