Principal Investigator: Martin Zeman
The proposed research focuses on various aspects of the inner model theory and can be divided into two areas. The first area puts emphasis on the determining the internal structure of extender models, fine structure theory and infinitary combinatorics. The PI has done extensive research here, making substantial contributions to the existing results. Extender models are generalizations of Goedel's constructible universe which admit a very complex structure of large cardinal axioms. The key element behind the inner model theory is iterability -- a method which enables us to iterate countable structures that are elementarily embeddable into initial segments of these models. Granting iterability, the author and his collaborators were able to develop a fine structure theory of these models, which is a means enabling to investigate the internal structure and combinatorics (Jensen's principles) of these models abstractly, without any direct reference to iterability. The aim of this project is to extend the existing methods in a manner that would yield the complete description of combinatorial properties of extender models. The open problems are related mainly to cardinal transfer theorems and stationary reflection. This part of the project includes also direct applications of the inner model theory in determining the consistency strength of various principles from infinitary combinatorics. The focus here is on improving the PI's results on Jensen's guessing principle. The second part of the project consists of two areas that are less tightly related to the PI's past research, namely the construction of inner models, iterability and applications in the descriptive set theory.
The subject of set theory is the analysis of methods which arise in mathematics. The methods that are widely accepted by the mathematical world have been formalized into so-called Zermelo-Fraenkel system of axioms (briefly ZFC). However, it turns out that we more and more often encouter questions whose solutions require more than mere ZFC. Set theory provides us with tools for recognizing such problems and approaches which enable us to determine which methods to use. More precisely, set theory provides us with a general method for determining the complexity of various problems quantitatively. The scale used here is the hierarchy of the large cardinal axioms. Extender models play a crucial role here -- there are the actual technical means which enables us to establish the connection between various concrete problems from mathematics and large cardinal axioms.
The proposed research focuses on various aspects of the inner model theory and can be divided into two areas. The first area puts emphasis on the determining the internal structure of extender models, fine structure theory and infinitary combinatorics. The PI has done extensive research here, making substantial contributions to the existing results. Extender models are generalizations of Goedel's constructible universe which admit a very complex structure of large cardinal axioms. The key element behind the inner model theory is iterability -- a method which enables us to iterate countable structures that are elementarily embeddable into initial segments of these models. Granting iterability, the author and his collaborators were able to develop a fine structure theory of these models, which is a means enabling to investigate the internal structure and combinatorics (Jensen's principles) of these models abstractly, without any direct reference to iterability. The aim of this project is to extend the existing methods in a manner that would yield the complete description of combinatorial properties of extender models. The open problems are related mainly to cardinal transfer theorems and stationary reflection. This part of the project includes also direct applications of the inner model theory in determining the consistency strength of various principles from infinitary combinatorics. The focus here is on improving the PI's results on Jensen's guessing principle. The second part of the project consists of two areas that are less tightly related to the PI's past research, namely the construction of inner models, iterability and applications in the descriptive set theory.
The subject of set theory is the analysis of methods which arise in mathematics. The methods that are widely accepted by the mathematical world have been formalized into so-called Zermelo-Fraenkel system of axioms (briefly ZFC). However, it turns out that we more and more often encouter questions whose solutions require more than mere ZFC. Set theory provides us with tools for recognizing such problems and approaches which enable us to determine which methods to use. More precisely, set theory provides us with a general method for determining the complexity of various problems quantitatively. The scale used here is the hierarchy of the large cardinal axioms. Extender models play a crucial role here -- there are the actual technical means which enables us to establish the connection between various concrete problems from mathematics and large cardinal axioms.
