ABSTRACT

My project focuses on studying connections between inner models
of set theory and large cardinals on the one side, and infinitary
combinatorics and descriptive set theory on the other side. It
consists of four parts that are connected through methods of
research. The first part is devoted to the analysis of the internal
structure and combinatorial properties of extender models, an area
of research launched by Jensen in the late 60's; the emphasis here
is on extending the methods recently developed by Schimmerling and
Zeman, and creating a "catalogue" of such methods with broad
applicability. The second area focuses on optimal forcing
constructions in infinite combinatorics. A typical question in
this category is obtaining the exact consistency strength for
the failure of Jensen's principle "square" at a singular cardinal.
Although the emphasis here is on developing new forcing methods,
inner model theory (even at its current state of development)
provides large cardinal axioms that are likely candidates for the
consistency strengths in question. The third area comprises
applications of inner models in establishing lower bounds for
consistency strengths at large cardinal levels below one Woodin
cardinal, as well as applications in descriptive set theory
related to correctness of canonical inner models and inner model
theoretic characterizations of descriptive set theoretic objects.
The last area focuses on the theory of inner models; the main
objective here is to make a progress on extender models matching
higher levels of the large cardinal hierarchy, as well as
exploring possibilities for constructions of the core model below
one Woodin cardinal with no background large cardinal assumption
on the set-theoretic universe.

Set theory can be viewed as a mathematical theory that formalizes the
methods currently accepted as valid working methods in mathematics.
It is based on the so-called Zermelo-Fraenkel axioms that describe
basic mathematical constructions. The developments in mathematics
in the twentieth century brought the entire subject to a new stage:
It turns out that there are more and more natural questions answers
of which are sensitive to the background axioms, that is to the axioms
of set theory. Such questions arise even in classical disciplines like
algebra or analysis. They cannot be decided from Zermelo-Fraenkel
axioms alone. In order to decide such questions, it is necessary
to augment the axiomatic system by additional axioms; such axioms
are usually formulated in the language of infinitary combinatorics.
This has to be done in a manner that will not introduce an
inconsistency of the augmented axiomatic system. In some cases,
the consistency of the augmented system (relative to the
Zermelo-Fraenkel system) can be proved using methods formalizable
in the Zermelo-Fraenkel system itself. However, there is an entire
hierarchy of axioms, the so-called large cardinal axioms, which do not
allow this. It is believed that every mathematical statement that is not
decidable from the Zermelo-Fraenkel axioms alone -- or at least the
consistency of such a statement -- can be decided using the right large
cardinal axiom. The mathematical praxis provides a lot of support for
this belief. Thus, the large cardinal hierarchy constitutes a kind of
a "scale" that "measures" the complexity of mathematical statements, and
each such statement has an exact match on this scale. The large
cardinal axiom, that together with Zermelo-Fraenkel axioms provides
the very piece of information necessary and sufficient to decide the
(consistency of) the statement, is called the consistency
strength of the statement. Expressed informally: By determining the
consistency strengths, set theory is able to isolate the precise amount
of information that that has to be used along with the standard
mathematical methods in order to decide certain mathematical questions.
Inner model theory, forcing, descriptive set theory, and infinitary
combinatorics constitute crucial tools for this task.

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