Regularity Properties and Determinacy
Yurii Khomskii
Abstract:
One of the most intriguing developments of modern set theory is the
investigation of two-player infinite games of perfect information. Of
course, it is clear that applied game theory, as any other branch of
mathematics, can be modeled in set theory. But we are talking about
the converse: the use of infinite games as a tool to study fundamental
set theoretic questions. When such infinite games are played using
integers as moves, a surprisingly rich theory appears, with
connections and consequences in all fields of pure set theory,
particularly the study of the continuum (the real numbers) and
Descriptive Set Theory (the study of "definable" sets of reals).
The concept of determinacy of games-a game is determined if one of the
players has a winning strategy-plays a key role in this field. In the
1960s, the Polish mathematicians Jan Mycielski and Hugo Steinhaus
proposed the famous Axiom of Determinacy (AD), which implies that all
sets of reals are Lebesgue measurable, have the Baire property, the
Perfect Set Property, and in general all the "regularity
properties". This contradicts the Axiom of Choice (AC) which allows us
to construct irregular sets by using an enumeration of the
continuum. A lot of work on determinacy is therefore done in ZF, i.e.,
Zermelo-Fraenkel set theory without the Axiom of Choice. In such a
mathematical universe with AC replaced by AD, the pathological,
nonconstructive sets that form counter-examples to the regularity
properties are altogether banished.
But how should we understand determinacy in the context of ZFC, i.e.,
standard Zermelo-Fraenkel set theory with Choice? The easiest way is
to look at determinacy as another kind of regularity property, D,
where a set of reals A is determined if its corresponding game is
determined. Since in the AD context infinite games are used to prove
regularities, one would expect determinacy to be a kind of "mother
regularity property", one which subsumes and implies all the
others. This is indeed true, but only in the "classwise" sense:
assuming for some large collection Gamma of sets that each of them is
determined, we may conclude that each set in Gamma has the regularity
properties. Does determinacy actually have "pointwise" consequences,
i.e., if we know of a set A that it is determined, does that imply
that A is regular? In general, the answer is no. The real "mother
regularity property" is the much stronger property of being
homogeneously Suslin, which does imply all the regularity properties
pointwise.1 Although there are close similarities between determinacy
and being homogeneously Suslin, the crucial difference lies in the
fact that the former has only classwise consequences whereas the
latter has pointwise consequences. In this sense determinacy is a
relatively weak property.
Although, from the beginning, researchers were aware of this fact, a
rigorous study of pointwise (non-)implications from determinacy has
not been carried out until a paper by Loewe in 2005. In this thesis,
we will continue the research started in that paper and generalize
some of its results.
Another focus of this thesis are the regularity properties
themselves. We take the view that most regularity properties are
naturally connected with special combinatorial objects called forcing
partial orders. The motivation comes from the theory of forcing, a
mainstream area dealing with the independence of certain propositions
(like the Continuum Hypothesis) from the axioms of set theory. These
combinatorial objects are also interesting in their own right, and can
be put in connection with classical regularity properties (e.g., the
Baire property and the Perfect Set Property) as well as other
regularity properties. There are still a number of open questions
regarding these connections.
This thesis will combine the study of pointwise consequences of
determinacy with the study of these general open questions.
Concretely, we denote a particular forcing partial order by P. Some P
generate a topology, whereas others don't, and this distinction into
topological versus non-topological forcing notions will be central to
our work. The most important regularity property connected to P is the
Marczewski-Burstin algebra denoted by MB(P), which can easily be
defined for any P. However, when P is topological, this algebra tends
to be a "bad" regularity property and is replaced by the Baire
property in the topology generated by P, denoted by BP(P). But this is
only a heuristic distinction, and no research has yet been done on
what the precise reason for the dichotomy is. This leads us to
formulate our first research question:
Main Question 1: Why is there a dichotomy between topological and
nontopological forcings P, i.e., why is it that for non-topological
forcings P the right regularity property is MB(P) whereas for
topological ones it is BP(P)? When is MB(P) a "good" property, and
what is the relationship between the two regularity properties?
Moving on toward pointwise consequences of determinacy, we wish to
study the connections between determinacy and the regularity
properties introduced above. In Loewe's paper, the case of
non-topological forcings P and the corresponding algebras MB(P) is
covered, where it is proved that in all interesting cases determinacy
does not imply MB(P) pointwise. Also, a weak version of the
Marczewski-Burstin algebra, denoted by wMB(P), is introduced and
studied (where the connections with determinacy are more
interesting). We will do an analogous analysis for the topological
case.
Main Question 2: Can we do an analysis of the pointwise connection
between determinacy and the Baire property BP(P) (for topological
P), similar to the one in Loewe's paper? Can we also introduce a
weak version of the Baire property wBP(P), and if so, what is the
pointwise connection between determinacy and wBP(P)?
If BP(P) was a generalization of the standard Baire property, then
there are also several generalizations of the Perfect Set
Property. These so-called asymmetric regularity properties can also be
connected to forcing partial orders P, in which case we denote them by
Asym(P). In current research, there are four particular examples but
as of yet no general definition. We would like to find that general
definition, and also to study the pointwise connections with
determinacy, analogously to Question 2. This leads us to the last
research question:
Main Question 3: Can a general definition for the asymmetric
property Asym(P) be given? If so, can we do a similar analysis for
the pointwise connections between determinacy and Asym(P) as we did
in Question 2?
This thesis is structured as follows: in Chapter 1, we introduce the
basic definitions and ideas related to the study of the real numbers
and the forcing notions. Chapter 2 is still introductory, developing
in detail the key ideas: determinacy, regularity properties, pointwise
and classwise implications. In Chapter 3 we deal with Main Question
1. The main result there is Theorem 3.4 which provides the connection
between MB and BP. In the rest of the chapter we study other aspects
of Question 1 (when is MB(P) a \sigma-algebra) and provide a partial
answer in Theorems 3.6 and Theorem 3.13.
In Chapter 4 we deal with Main Question 2. Analogously to Loewe's
paper we prove that determinacy does not imply BP(P) pointwise
(Theorem 4.8) and characterize the P for which determinacy does, or
does not, imply the weak Baire property pointwise (Theorems 4.13 and
4.18).
Finally, in Chapter 5 we deal with Main Question 3. Although we do not
find a clear definition for Asym(P), we do give a necessary condition
which such a property must satisfy, in terms of a game
characterization. This characterization is sufficient to solve the
second part of the question: in Theorem 5.12 we do prove that
determinacy does not imply Asym(P) pointwise in all non-trivial cases.