DS201004: Ikegami, Daisuke (2010) Games in Set Theory and Logic. Doctoral thesis, University of Amsterdam.
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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 BanachMazur games. We characterize almost all
the known regularity properties for sets of reals via the Baire
property for some topological spaces and use BanachMazur 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 GaleStewart 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
GaleStewart 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 (BlAD_R). We show that the consistency strength of
BlAD_R is strictly greater than that of the Axiom of Determinacy (AD)
and that BlAD_R implies almost all the known regularity properties
for every set of reals. We discuss the possibility of the equivalence
between AD_R and BlAD_R under the ZermeloFraenkel set theory with
the Axiom of Dependent Choice (ZF+DC) and the possibility of the
equiconsistency between AD_R and BlAD_R.
Chapter 4: We work on the connection between the determinacy of
GaleStewart games and large cardinals. Iteration trees are important
objects to prove the determinacy of GaleStewart games from large
cardinals and alternating chains with length \omega are the most
fundamental iteration trees connected to the determinacy of
GaleStewart 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 manyone 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 firstorder 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 \mucalculus, and a fragment of the modal
\mucalculus are product closed.
Keywords:
Item Type:  Thesis (Doctoral) 

Report Nr:  DS201004 
Series Name:  ILLC Dissertation (DS) Series 
Year:  2010 
Subjects:  Logic 
Depositing User:  Dr Marco Vervoort 
Date Deposited:  14 Jun 2022 15:16 
Last Modified:  14 Jun 2022 15:16 
URI:  https://eprints.illc.uva.nl/id/eprint/2087 
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