MoL-2007-11:
Khomskii, Yurii
(2007)
*Regularity Properties and Determinacy.*
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## 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.

Item Type: | Report |
---|---|

Report Nr: | MoL-2007-11 |

Series Name: | Master of Logic Thesis (MoL) Series |

Year: | 2007 |

Date Deposited: | 12 Oct 2016 14:38 |

Last Modified: | 12 Oct 2016 14:38 |

URI: | https://eprints.illc.uva.nl/id/eprint/783 |

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