This volume contains the basics of Zermelo-Fraenkel axiomatic set theory. It is situated between two opposite poles: On one hand there are elementary texts that familiarize the reader with the vocabulary of set theory and build set-theoretic tools for use in courses in analysis, topology, or algebra – but do not get into metamathematical issues. On the other hand are those texts that explore issues of current research interest, developing and applying tools (constructibility, absoluteness, forcing, etc.) that are aimed to analyze the inability of the axioms to settle certain set-theoretic questions.

Much of this volume just “does set theory”, thoroughly developing the theory of ordinals and cardinals along with their arithmetic, incorporating a careful discussion of diagonalization and a thorough exposition of induction and inductive (recursive) definitions. Thus it serves well those who simply want tools to apply to other branches of mathematics or mathematical sciences in general (e.g., theoretical computer science), but alsowant to find out about some of the subtler results of modern set theory.

Moreover, a fair amount is included towards preparing the advanced reader to read the research literature. For example, we pay two visits to Gödel's constructible universe, the second of which concludes with a proof of the relative consistency of the axiom of choice and of the generalized continuum hypothesis with ZF. As such a program requires, I also include a thorough discussion of formal interpretations and absoluteness. The lectures conclude with a short but detailed study of Cohen forcing and a proof of the non-provability in ZF of the continuum hypothesis.

In Chapter VI, among other things, we studied the WO sets and learnt how to measure their length with the help of ordinal numbers. A consequence of the axiom of choice was (Theorem VI.5.50) that every set can be well-ordered and therefore every set can be assigned a length.

In the present chapter we turn to another aspect of set size, namely its number of elements, or cardinality. It will turn out that for finite sets length and cardinality are measured by the same (finite) ordinal; thus, in particular, finite sets have a unique length. As was already remarked, the situation with infinite sets is much less clean intuitively, and several WO sets of differing lengths can have the same number of elements (e.g., ω, ω + 1, ω + 2, etc).

The following section will formalize the notions of “finite” and “infinite” sets. Intuitively, a set is finite if the process of removing its elements, one at a time, will terminate; it is infinite otherwise.

Thus for finite sets the process implicitly assigns the numbers 1, 2, 3,… to the first, second, third, … removed items. Since the process terminates, there will be a natural number assigned to the last removed item. Evidently this number equals the cardinality, or number of elements, of the set.

In the infinite case it is not clear a priori how to assign a “number” that denotes the cardinality of the set. Thus the issue is temporarily postponed, and one first worries about whether or not two infinite sets have the same number of elements.

The method of forcing was invented by Cohen (1963) towards the construction of non-standard models of ZFC, so that “new axioms” could be proved consistent with the standard ones. Our retelling of the basics of forcing found in this chapter is indebted primarily to the user-friendly account found in Shoenfield (1971). The influence of the expositions in Burgess (1978), Jech (1978b), and Kunen (1980) should also be evident.

In outline, the method goes like this: Suppose we want to show that ZFC (sometimes ZF or an even weaker subtheory) is consistent with some weird new axiom, “NA”. Working in the metatheory, one starts with a CTM, M, for ZFC. This is the ground model. One then judiciously chooses a PO set, 〈P, <, 1〉, in M – where we find it convenient to restrict attention to PO sets that have a maximum element (let us call the latter “1”) – and, using the PO set, one constructs a so-called generic set G. Circumstances normally have G obey G ∉ M. The “judicious” aspect of the choice of the PO set will entail that the generic extension, M[G], of the CTM M not only contains G as an element but is a CTM itself that satisfies NA as well (i.e., ⊨M[G] ZFC+NA). Thus, one has a proof in the metatheory that if ZFC is consistent (i.e., if a CTM for ZFC exists), then so is ZFC + NA.

We have said above that “〈P,<, 1〉 ∈ M”. By absoluteness of pair (see Section VI.8), the quoted statement is equivalent to “P ∈ M and<∈ M and 1 ∈ M”.

This prerequisite chapter – what some authors call a “Chapter 0” – is an abridged version of Chapter I of volume 1 of my Lectures in Logic and Set Theory. It is offered here just in case that volume Mathematical Logic is not readily accessible.

Simply put, logic is about proofs or deductions. From the point of view of the user of the subject – whose best interests we attempt to serve in this chapter – logic ought to be just a toolbox which one can employ to prove theorems, for example, in set theory, algebra, topology, theoretical computer science, etc.

The volume at hand is about an important specimen of a mathematical theory, or logical theory, namely, axiomatic set theory. Another significant example, which we do not study here, is arithmetic. Roughly speaking, a mathematical theory consists on one hand of assumptions that are specific to the subject matter – the so-called axioms – and on the other hand a toolbox of logical rules. One usually performs either of the following two activities with a mathematical theory: One may choose to work within the theory, that is, employ the tools and the axioms for the sole purpose of proving theorems. Or one can take the entire theory as an object of study and study it “from the outside” as it were, in order to pose and attempt to answer questions about the power of the theory (e.g., “does the theory have as theorems all the ‘true’ statements about the subject matter?”), its reliability (meaning whether it is free from contradictions or not), how its reliability is affected if you add new assumptions (axioms), etc.

This volume is an introduction to formal (axiomatic) set theory. Putting first things first, we are attempting in this chapter to gain an intuitive understanding of the “real” universe of sets and the process of set creation (that is, what we think is going on in the metatheory). After all, we must have some idea of what it is that we are called upon to codify and formally describe before we embark upon doing it.

Set theory, using as primitives the notions of set (as a synonym for “collection”), atom (i.e., an object that is not subdivisible, not a collection), and the relation belongs to (∈), has sufficient expressive power to serve as the foundation of all mathematics. Mathematicians use notation and results from set theory in their everyday practice. We call the sets that mathematicians use the “real sets” of our mathematical intuition.

The exposition style in this chapter, true to the attribute “naïve”, will be rather leisurely to the extent that we will forget, on occasion, that our “Chapter 0” (Chapter I) is present.

The “Real Sets”

Naïvely, or informally, set theory is the study of collections of “mathematical objects”.

Informal Description (Mathematical Objects). Set theory is only interested in mathematical objects. As far as set theory is concerned, such objects

are either

1. atomic – let us understand by this term an object that is not a collection of other objects – such as a number or a point on a Euclidean line, or

2. collections of mathematical objects.