The course presented in this text concentrates on the typical methods of modern set theory: transfinite induction, Zorn's lemma, the continuum hypothesis, Martin's axiom, the diamond principle ◊, and elements of forcing. The choice of the topics and the way in which they are presented is subordinate to one purpose – to get the tools that are most useful in applications, especially in abstract geometry, analysis, topology, and algebra. In particular, most of the methods presented in this course are accompanied by many applications in abstract geometry, real analysis, and, in a few cases, topology and algebra. Thus the text is dedicated to all readers that would like to apply set-theoretic methods outside set theory.

The course is presented as a textbook that is appropriate for either a lower-level graduate course or an advanced undergraduate course. However, the potential readership should also include mathematicians whose expertise lies outside set theory but who would like to learn more about modern set-theoretic techniques that might be applicable in their field.

The reader of this text is assumed to have a good understanding of abstract proving techniques, and of the basic geometric and topological structure of the n-dimensional Euclidean space ℝn. In particular, a comfort in dealing with the continuous functions from ℝn into ℝ is assumed. A basic set-theoretic knowledge is also required.

Our main prerequisites may be listed as follows.

1.1Elementary theory of Lie groups and Lie algebras, and the interpretation of the elements of the Lie algebra as differential operators on the group or its coset spaces. This will be used only for SL2(ℝ), its Lie subgroups, and the upper halfplane. The book by Warner [58] is more than sufficient for our needs, except for some facts on Haar measures (2.9, 10.9).

1.2The regularity theorem for elliptic operators (see the remark in 2.13 for references).

1.3Some functional analysis, mainly about operators on Hilbert spaces. For the sake of definiteness, I have used two basic textbooks ([46] and [51]) and have given at least one precise reference for every theorem used. But this material is standard and the reader is likely to find what is needed in his (or her) favorite book on functional analysis. The demands will increase as we go along, and the material will be briefly reviewed before it is needed. Such review is mainly intended to refresh memory, fix notation, and give references, not to be a full-fledged introduction.

1.4Infinite dimensional representations of G. We shall review what we need in Sections 14 and 15 and also give a few proofs, but mostly refer to the literature. An essentially self-contained discussion of some of the results on SL2(ℝ) stated in Sections 2 and 14 is contained in the first part of [40].

From the results of Section 1.2 it is clear that sets such as ∅, {∅}, {∅, {∅}}, {{∅}}, and so forth exist. Using the axiom of infinity we can also conclude that we can build similar infinite sets. But how do we construct complicated sets, such as the sets of natural and real numbers, if the only tools we have to build them are the empty set ∅ and “braces” {·}? We will solve this problem by identifying the aforementioned objects with some specific sets.

Briefly, we will identify the number 0 with the empty set ∅, and the other natural numbers will be constructed by induction, identifying n with {0, …, n − 1}. The existence of the set ℕ of all natural numbers is guaranteed by the infinity axiom. The real numbers from the interval [0,1] will be identified with the set of functions {0,1}ℕ, where an infinite sequence a: ℕ → {0,1} is identified with the binary expansion of a number, that is, with Σn∈ℕa(n)/2n+1. The details of these constructions are described in the rest of this chapter.

Natural numbers

In this section we will find a set that represents the set ℕ of natural numbers in our set-theoretic universe. For this, we need to find for each natural number n a set that represents it. Moreover, we will have to show that the class of all such defined natural numbers forms a set.

Why axiomatic set theory?

Essentially all mathematical theories deal with sets in one way or another. In most cases, however, the use of set theory is limited to its basics: elementary operations on sets, fundamental facts about functions, and, in some cases, rudimentary elements of cardinal arithmetic. This basic part of set theory is very intuitive and can be developed using only our “good” intuition for what sets are. The theory of sets developed in that way is called “naive” set theory, as opposed to “axiomatic” set theory, where all properties of sets are deduced from a fixed set of axioms.

Clearly the “naive” approach is very appealing. It allows us to prove a lot of facts on sets in a quick and convincing way. Also, this was the way the first mathematicians studied sets, including Georg Cantor, a “father of set theory.” However, modern set theory departed from the “paradise” of this simple-minded approach, replacing it with “axiomatic set theory,” the highly structured form of set theory. What was the reason for such a replacement?

Intuitively, a set is any collection of all elements that satisfy a certain given property. Thus, the following axiom schema of comprehension, due to Frege (1893), seems to be very intuitive.

A more accurate title would be: Introduction to some aspects of the analytic theory of automorphic forms on SL2(ℝ) and the upper half-plane X. Originally, automorphic forms were holomorphic or meromorphic functions on X satisfying certain conditions with respect to a discrete group Γ of automorphisms of X, usually with fundamental domain of finite (hyperbolic) area. Later on, H. Maass – and then A. Selberg and W. Roelcke – dropped the assumption of holomorphicity, requiring instead that the functions under consideration be eigenfunctions of the Laplace–Beltrami operator. In the 1950s it was realized (in more general cases) – initially by I. M. Gelfand and S. V. Fomin, and then by Harish-Chandra – that the automorphic forms (holomorphic or not) could be equivalently viewed as functions on Γ\SL2(ℝ) satisfying certain conditions familiar in the theory of infinite dimensional representations of semisimple Lie groups. This led to a new outlook, where the Laplace–Beltrami operator is replaced by the Casimir operator and the theory of automorphic forms becomes closely related to harmonic analysis on Γ\SL2(ℝ). This is the point of view adopted in this presentation. However, in order to limit the prerequisites, no knowledge of representation theory is assumed until the last sections, a main purpose of which is precisely to make this connection explicit. A fundamental role is played throughout by a theorem stating that a function on SL2(ℝ) satisfying certain assumptions (Z-finite and K-finite) is fixed under convolution by some smooth function with arbitrarily small compact support around the identity element (2.14).

Strange subsets of ℝn and the diagonalization argument

In this section we will illustrate some typical constructions by transfinite induction. We will do so by constructing recursively some subsets of ℝn with strange geometric properties. The choice of geometric descriptions of these sets is not completely arbitrary here – we still have not developed the basic facts concerning “nice” subsets of ℝn that are necessary for most of our applications. This will be done in the remaining sections of this chapter.

To state the next theorem, we will need the following notation. For a subset A of the plane ℝ2 the horizontal section of A generated by y ∈ ℝ (or, more precisely, its projection onto the first coordinate) will be denoted by Ay and defined as Ay = {x ∈ ℝ: 〈x,y〉 ∈ A}. The vertical section of A generated by x ∈ ℝ is defined by Ax = {y ∈ ℝ: 〈x,y〉 ∈ A}.

We will start with the following example.

Theorem 6.1.1There exists a subset A of the plane with every horizontal section Ay being dense in ℝ and with every vertical section Ax having precisely one element.

Proof We will define the desired set by induction. To do so, we will first reduce our problem to the form that is most appropriate for a recursive construction.

The requirement that every horizontal section of A is dense in ℝ tells us that A is “reasonably big.”