Games in Set Theory and Logic
Daisuke Ikegam
Abstract:
In this dissertation, we discuss several types of infinite games and
related topics in set theory and mathematical logic. Chapter 1 is
devoted to the general introduction and preliminaries. The rest is
organized as follows:
Chapter 2: It is known that the Baire property is one of the nice
properties for sets of reals called regularity properties and that it
can be characterized by Banach-Mazur games. We characterize almost all
the known regularity properties for sets of reals via the Baire
property for some topological spaces and use Banach-Mazur games to
prove the general equivalence theorems between regularity properties,
forcing absoluteness, and the transcendence properties over some
canonical inner models. With the help of these equivalence results, we
answer some open questions from set theory of the reals.
Chapter 3: We discuss the connection between Gale-Stewart games and
Blackwell games where the former are infinite games with perfect
information coming from set theory and the latter are infinite games
with imperfect information coming from game theory. The determinacy of
Gale-Stewart games has been one of the main topics in set theory and
one could also consider the determinacy of Blackwell games. We compare
the Axiom of Real Determinacy (AD_R) and the Axiom of Real Blackwell
Determinacy (Bl-AD_R). We show that the consistency strength of
Bl-AD_R is strictly greater than that of the Axiom of Determinacy (AD)
and that Bl-AD_R implies almost all the known regularity properties
for every set of reals. We discuss the possibility of the equivalence
between AD_R and Bl-AD_R under the Zermelo-Fraenkel set theory with
the Axiom of Dependent Choice (ZF+DC) and the possibility of the
equiconsistency between AD_R and Bl-AD_R.
Chapter 4: We work on the connection between the determinacy of
Gale-Stewart games and large cardinals. Iteration trees are important
objects to prove the determinacy of Gale-Stewart games from large
cardinals and alternating chains with length \omega are the most
fundamental iteration trees connected to the determinacy of
Gale-Stewart games. We investigate the the upper bound of the
consistency strength of the existence of alternating chains with
length \omega.
Chapter 5: Wadge reducibility measures the complexity of subsets of
topological spaces via the continuous reduction of a subset of a
topological space to another one in descriptive set theory
corresponding to many-one reducibility in recursion theory. With the
help of the characterization of the Wadge reducibility for the Baire
space in terms of Wadge games, one can develop the beautiful theory of
the Wadge reducibility for the Baire space (e.g., almost linearity,
wellfoundedness) assuming the Axiom of Determinacy (AD). We study the
Wadge reducibility for the real line which cannot be characterized by
infinite games in a similar way. We show that the Wadge Lemma for the
real line fails and that the Wadge order for the real line is
illfounded and investigate more properties of the Wadge order for the
real line.
Chapter 6: Modal fixed point logics are modal logics with fixed point
operators and they enjoy several nice properties as first-order logic
has. We define a product construction of an event model and a Kripke
model and discuss the product closure of modal fixed point logics. We
show that PDL, the modal \mu-calculus, and a fragment of the modal
\mu-calculus are product closed.
Keywords: