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.