Introduction

Set theory begins with one of the simplest mathematical ideas: some things may be collected together and regarded as one object. The collection of prime numbers less than \(10\),

\[ \{2,3,5,7\}, \]

is itself a mathematical object. Its members are \(2\), \(3\), \(5\), and \(7\). The symbol

\[ x \in A \]

means that \(x\) is an element, or member, of the set \(A\). Thus \(3 \in \{2,3,5,7\}\), while \(4 \notin \{2,3,5,7\}\).

At first sight, this seems too simple to need a theory. Yet modern mathematics repeatedly turns collections into objects: the set of all real numbers, the set of all continuous functions from \(\mathbb{R}\) to \(\mathbb{R}\), the set of all open subsets of a topological space, the set of all ideals of a ring, the set of all countable ordinals. Once collections may themselves be collected, the idea becomes powerful enough to organize almost all ordinary mathematical structures. It also becomes subtle enough to produce paradoxes if used without discipline.

A basic example is Russell’s paradox. Suppose, naively, that every property determines a set. Then there should be a set

\[ R=\{x : x \notin x\}, \]

the set of all sets that are not members of themselves. Ask whether \(R \in R\). If \(R \in R\), then by the defining property of \(R\), we must have \(R \notin R\). If \(R \notin R\), then \(R\) satisfies the defining property, so \(R \in R\). Both alternatives contradict themselves. The lesson is not that sets are unusable; the lesson is that the principle “every property determines a set” is too strong.

Axiomatic set theory is the disciplined replacement. Instead of saying that every describable collection is automatically a set, we state precise axioms explaining which set-forming operations are allowed. The standard axiomatic system studied in this book is ZFC, short for Zermelo–Fraenkel set theory with the Axiom of Choice. Zermelo’s 1908 axiomatization was a decisive step toward replacing unrestricted set formation with controlled axioms, and later refinements led to the system now called ZFC (Zermelo 1908; Jech 2003). This book studies ZFC first as a formal foundation and then as the starting point for independence methods, especially forcing.

What set theory is trying to do

Set theory has two closely related tasks.

The first task is foundational. It provides a common language in which mathematical objects can be represented. Natural numbers may be represented as particular sets, ordered pairs may be represented as sets, functions may be represented as sets of ordered pairs, and spaces, groups, rings, fields, and measures may all be coded inside the universe of sets. The point is not that a working algebraist must constantly think of a group as a set-theoretic code. The point is that there is a uniform background theory in which such objects can be discussed and compared.

The second task is mathematical in its own right. Infinite sets have structure. There are different sizes of infinity, called cardinalities. There are canonical well-ordered objects, called ordinals, that measure order type and support transfinite induction. There are hierarchies of definability, large cardinal axioms, inner models, forcing extensions, and independence phenomena. Set theory is not merely bookkeeping for other areas; it is a field with its own theorems, constructions, and open problems.

A first example of genuine set-theoretic mathematics is Cantor’s comparison of infinite sizes. Two sets \(A\) and \(B\) have the same cardinality if there is a bijection between them, that is, a function \(f:A\to B\) that is both one-to-one and onto. The natural numbers

\[ \mathbb{N}=\{0,1,2,3,\dots\} \]

and the even natural numbers

\[ E=\{0,2,4,6,\dots\} \]

have the same cardinality because the function \(n\mapsto 2n\) is a bijection from \(\mathbb{N}\) to \(E\). This is one of the first surprises of infinite set theory: a proper subset of an infinite set can have the same size as the whole set.

But not all infinite sets have the same size. Cantor’s theorem says that for every set \(A\), the power set \(\mathcal{P}(A)\), the set of all subsets of \(A\), has strictly larger cardinality than \(A\). For example, the set of all subsets of \(\mathbb{N}\) is larger than \(\mathbb{N}\) itself. This result is central to the development of cardinal arithmetic and to the continuum problem treated later in the book (Jech 2003).

From axioms to independence

A central theme of this book is that axioms have consequences, but they also have limits.

A formal axiom system consists of a formal language, rules of logical inference, and selected statements called axioms. In first-order set theory, the language is especially spare: it has equality \(=\), the membership relation \(\in\), logical symbols such as \(\forall\), \(\exists\), \(\neg\), \(\wedge\), and variables ranging over sets. ZFC is expressed in this language.

For example, the Axiom of Extensionality says that two sets are equal exactly when they have the same elements:

\[ \forall A\,\forall B\bigl[(\forall x(x\in A \leftrightarrow x\in B))\rightarrow A=B\bigr]. \]

This axiom captures the idea that a set is determined by its members. The sets \(\{1,2,3\}\) and \(\{3,2,1\}\) are the same set because they have exactly the same elements. Order and repetition do not matter for sets: \(\{1,1,2,3\}=\{1,2,3\}\).

Other axioms permit controlled construction. Pairing says that from \(a\) and \(b\) we may form \(\{a,b\}\). Union says that from a set \(A\) we may form \(\bigcup A\), the set of all elements of elements of \(A\). Power Set says that from \(A\) we may form \(\mathcal{P}(A)\). Separation says that from an already existing set \(A\), we may carve out the subset of elements satisfying a given formula. Crucially, Separation does not say that every formula defines a set absolutely; it says that a formula may define a subset of a set already available.

This distinction is one of the basic habits of axiomatic set theory. The collection

\[ \{x\in \mathbb{N}: x \text{ is prime}\} \]

is a set by Separation, because it is obtained by selecting certain elements from the already existing set \(\mathbb{N}\). By contrast, the collection of all sets is not a set in ZFC. If there were a set \(V\) containing every set, then \(\mathcal{P}(V)\) would also be a set, and Cantor’s theorem would give \(|\mathcal{P}(V)|>|V|\), although every element of \(\mathcal{P}(V)\) would already be a set and hence an element of \(V\). The universe of all sets is therefore treated as a proper class, not as one more set.

An independence result shows that a statement cannot be proved or refuted from a given axiom system, assuming the system is consistent. More precisely, a sentence \(\varphi\) is independent of a theory \(T\) if neither \(\varphi\) nor \(\neg\varphi\) is provable from \(T\). In set theory, the most famous example is the continuum hypothesis.

The continuum hypothesis, abbreviated CH, concerns the size of the real line. The set \(\mathbb{R}\) has the same cardinality as \(\mathcal{P}(\mathbb{N})\), often denoted \(2^{\aleph_0}\). CH states that there is no cardinal strictly between the cardinality of \(\mathbb{N}\), denoted \(\aleph_0\), and the cardinality of \(\mathbb{R}\):

\[ \text{CH:}\qquad 2^{\aleph_0}=\aleph_1. \]

Here \(\aleph_1\) is the least uncountable cardinal. Informally, CH says that the real line is the smallest possible uncountable size.

Gödel showed that if ZF is consistent, then ZF together with the Axiom of Choice and the generalized continuum hypothesis is consistent, by constructing the inner model \(L\), the constructible universe (Gödel 1940). Cohen later introduced forcing and used it to prove, among other results, that CH cannot be proved from ZFC if ZFC is consistent; together with Gödel’s result, this yields the independence of CH from ZFC relative to the consistency of ZFC (Cohen 1966; Kunen 1980). This is one of the great turning points of twentieth-century logic: a natural question about the size of the continuum is not settled by the standard axioms alone.

Models, universes, and the meaning of relative consistency

To understand independence, we must learn to reason about models of set theory. A model is a mathematical structure in which the symbols of a language receive interpretations and the axioms become true statements inside that structure.

For the language of set theory, a model consists of a domain \(M\) together with a binary relation \(E\) on \(M\), where \(E\) interprets membership. We write

\[ (M,E)\models \varphi \]

to mean that the sentence \(\varphi\) is true in the structure \((M,E)\). If \(E\) is the actual membership relation restricted to \(M\), one often writes simply \(M\models\varphi\), provided the context is clear.

A particularly important kind of model is a transitive model. A set \(M\) is transitive if whenever \(x\in y\in M\), then \(x\in M\). In a transitive model, membership among elements of \(M\) agrees well with ordinary membership. For example, if \(M\) is transitive and \(y\in M\), then every element of \(y\) is also in \(M\). This makes transitive models technically convenient, especially in the study of absoluteness and forcing.

Relative consistency arguments have the following general shape. Suppose we can show that from any model of a theory \(T\), we can construct a model of \(T+\varphi\). Then, if \(T\) has a model, so does \(T+\varphi\). By Gödel’s completeness theorem, this gives a relative consistency result: if \(T\) is consistent, then \(T+\varphi\) is consistent. In set theory, this method is delicate because one must control which axioms remain true in the constructed model. Inner models such as \(L\) and outer models produced by forcing are the main tools developed for this purpose (Jech 2003; Kunen 1980).

The word relative matters. A relative consistency proof does not prove that ZFC is absolutely consistent. Gödel’s second incompleteness theorem implies that, under suitable hypotheses, a sufficiently strong consistent theory cannot prove its own consistency. Instead, set theorists compare theories: if one theory is consistent, then another is consistent. This comparison leads naturally to the study of consistency strength, especially in connection with large cardinal axioms (Kanamori 2009).

Forcing in first orientation

Forcing is the central method by which this book moves from ZFC to modern independence results. At a first approximation, forcing constructs a larger universe of sets by adding a new object in a controlled way. The control is supplied by a partially ordered set, or poset, whose elements are called conditions.

A poset is a set \(P\) with a relation \(\leq\) that is reflexive, transitive, and antisymmetric. In forcing, the convention is often that \(q\leq p\) means “\(q\) is stronger than \(p\).” A stronger condition gives more information.

The simplest guiding example is Cohen forcing for adding a new binary sequence. Let

\[ P=2^{<\omega}, \]

the set of all finite sequences of \(0\)s and \(1\)s. A condition is a finite approximation to an infinite binary sequence. If \(p=(1,0,1)\), then \(p\) says: the first bit is \(1\), the second bit is \(0\), and the third bit is \(1\). A longer sequence extending \(p\), such as \(q=(1,0,1,1,0)\), gives more information, so \(q\leq p\).

A dense set is a set of conditions that cannot be avoided: below every condition, there is a stronger condition in the dense set. For example, for each natural number \(n\), the set of finite binary sequences of length greater than \(n\) is dense in \(2^{<\omega}\), because any finite sequence can be extended until it has length greater than \(n\). A generic filter is a coherent family of conditions that meets all dense sets of a specified model. In this example, meeting the dense set for each \(n\) ensures that the finite approximations determine an infinite binary sequence.

This informal description hides serious work. We must define names for objects in the extension, prove the forcing theorem, show that the extension satisfies the axioms of set theory, and analyze which cardinals, cofinalities, and combinatorial properties are preserved. These tasks occupy Chapters 12 through 15. But the central intuition is already visible: forcing adds information while preserving enough structure to keep set theory under rigorous control.

The route through the book

The book follows a deliberately cumulative path.

Chapters 1 through 4 build the core language of axiomatic set theory. We begin with the axioms of ZFC, then develop classes, relations, functions, well-orderings, ordinals, cardinals, cofinality, and cardinal arithmetic. These chapters establish the technical vocabulary needed throughout the rest of the book.

Chapters 5 through 7 organize the universe internally. The cumulative hierarchy \(V_\alpha\) explains how sets are built in stages. Rank measures the stage at which a set appears. Choice principles are compared through equivalent formulations such as Zorn’s lemma and the well-ordering theorem. Filters, ideals, and Boolean algebras introduce algebraic structures that later reappear in forcing.

Chapters 8 and 9 introduce models, satisfaction, elementary substructures, the Mostowski collapse, absoluteness, and reflection. These chapters teach a new habit: instead of reasoning only inside one universe, we compare what different models see.

Chapter 10 constructs Gödel’s universe \(L\). This is the first major inner model of the book. It shows how definability can produce a canonical class model of ZFC and explains why both the Axiom of Choice and the generalized continuum hypothesis hold in \(L\) (Gödel 1940; Jech 2003).

Chapter 11 clarifies independence and relative consistency. It prepares the conceptual ground for forcing by explaining what independence results do and do not say.

Chapters 12 through 15 develop forcing systematically: posets, names, generic filters, generic extensions, the forcing relation, preservation theorems, classical forcing constructions, and iterated forcing. These chapters form the technical center of the book.

Chapters 16 through 19 explore major regions of modern set theory: stationary and club sets, trees, partition relations, large cardinals, descriptive set theory, regularity properties, Martin’s axiom, proper forcing, forcing axioms, ideals, and continuum patterns. Large cardinals, in particular, form a scale of strong axioms of infinity and play a major role in contemporary consistency-strength comparisons (Kanamori 2009).

Chapter 20 returns to foundations. Once one has seen inner models, forcing extensions, large cardinals, and alternative set-theoretic universes, foundational questions become sharper. Should ZFC be viewed as the final background theory, or as a central theory among many possible extensions? What makes a new axiom mathematically compelling? How should independence influence ordinary mathematics? The final chapter does not pretend that these questions have easy answers. It gives the technical background needed to understand why they are real.

How to work mathematically in this book

This is a graduate-level book. It assumes comfort with proofs, first-order logic at an introductory level, and standard undergraduate mathematics. But it does not assume that you already know forcing, large cardinals, or inner model theory.

The most important practice is to distinguish four kinds of statements.

A definition introduces terminology. For example, “a set \(A\) is transitive if every element of an element of \(A\) is itself an element of \(A\)” is a definition. A definition is not something to prove; it is something to understand and use accurately.

A theorem is a statement proved from earlier definitions, axioms, and theorems. Cantor’s theorem, which says that \(|A|<|\mathcal{P}(A)|\), is a theorem of ZFC.

An axiom is an accepted starting principle of a theory. The Axiom of Power Set is not proved from earlier ZFC axioms inside the usual presentation of ZFC; it is one of the principles being assumed.

An independence result is a metamathematical statement about what cannot be proved from a theory, assuming consistency. “CH is independent of ZFC” is not itself a statement of ordinary cardinal arithmetic inside ZFC. It is a statement about the deductive power of ZFC, established by comparing models.

When reading, pause whenever the level changes. If a paragraph moves from a definition to a theorem, ask what must be proved. If it moves from a theorem to a relative consistency result, ask what models are being constructed. If it moves from ZFC to an extension of ZFC, ask which additional axiom is being assumed and what it is meant to accomplish.

Set theory rewards precision. Small differences matter: a set versus a class, membership versus subset, countable versus finite, true in \(V\) versus true in a model \(M\), provable from ZFC versus consistent with ZFC. The purpose of this book is to make those distinctions familiar enough that they become tools rather than obstacles.

References

Cohen, Paul J. 1966. Set Theory and the Continuum Hypothesis. New York: W. A. Benjamin.

Gödel, Kurt. 1940. The Consistency of the Axiom of Choice and of the Generalized Continuum-Hypothesis with the Axioms of Set Theory. Annals of Mathematics Studies, No. 3. Princeton: Princeton University Press.

Jech, Thomas. 2003. Set Theory: The Third Millennium Edition, Revised and Expanded. Springer Monographs in Mathematics. Berlin: Springer.

Kanamori, Akihiro. 2009. The Higher Infinite: Large Cardinals in Set Theory from Their Beginnings. 2nd ed. Springer Monographs in Mathematics. Berlin: Springer.

Kunen, Kenneth. 1980. Set Theory: An Introduction to Independence Proofs. Studies in Logic and the Foundations of Mathematics, Vol. 102. Amsterdam: North-Holland.

Zermelo, Ernst. 1908. “Untersuchungen über die Grundlagen der Mengenlehre. I.” Mathematische Annalen 65: 261–281.