Set Theory I develops the core foundations of modern set theory, beginning with the ZFC axioms and the cumulative hierarchy and then building a working theory of infinite mathematical objects. The course treats sets as the basic language for mathematics and studies how they organize relations, functions, orders, ordinals, and cardinals. Its focus is not just on definitions, but on the structural principles that govern infinite collections and the methods used to prove results about them.

The main themes are transfinite methods, size and comparison of infinite sets, and the interaction between combinatorial, logical, and structural ideas. After introducing relations and well-orders, the course develops transfinite induction and recursion, ordinal arithmetic, and the theory of cardinals and equinumerosity. It then moves to the [Axiom of Choice](/page/Axiom%20of%20Choice) and its equivalents, cardinal arithmetic, cofinality, regularity, stationary sets, Fodor's lemma, and the basic machinery of transitive sets, collapse, absoluteness, reflection, and elementary submodels.

Each chapter builds on the preceding ones by extending the same foundational toolkit to a deeper level of generality. The early chapters establish the universe of sets and the language for measuring and ordering infinite objects; the middle chapters use that language to analyze ordinals, cardinals, and choice principles; and the later chapters move toward finer structural and logical properties of the set-theoretic universe. The final chapter introduces constructibility as a first inner model, giving a first systematic example of how set theory studies alternative universes of sets and the principles that hold within them.

# Introduction

This introductory chapter fixes the viewpoint of the course before the axioms begin. Set theory will be treated both as a formal first-order theory and as the common language in which ordinary mathematical objects are coded. The guiding tension is that we want sets to be flexible enough to support ordinary construction, while the axioms must block unrestricted comprehension.

## Why Axiomatise Sets?

The first problem is not how to list familiar sets, but how to say which set-forming operations are legitimate. Naive set theory suggests that any property should determine a collection, yet the Russell construction shows that unrestricted comprehension is inconsistent. ZFC replaces this by axioms that build sets from already-given sets and by schemas that carve definable subsets out of existing sets.

[explanation: The Foundational Role of ZFC] ZFC is a first-order theory in the language with equality and one non-logical binary relation symbol $\in$. Its intended objects are sets, and all mathematical structures are represented by sets equipped with additional relations and functions. A group, a topological space, or a model of a first-order theory is coded by ordered tuples, functions, relations, and underlying sets.

This coding convention does not erase mathematical practice. Instead, it supplies a common background theory strong enough to formalise constructions and compare their strength. Later courses in forcing and large cardinals study extensions or refinements of this background, so this course concentrates on the basic machinery that those subjects assume. [/explanation]

The contrast with naive comprehension is the first lesson of the course: collections may be discussed informally, but only some collections are sets. This distinction leads to the working use of classes.

[definition: Class] A class is a collection described by a first-order formula $\varphi(x, p_1, \dots, p_n)$ with set parameters $p_1, \dots, p_n$. [/definition]

A class need not be a set. In these notes, classes are metalinguistic devices for talking about definable collections; they are not additional objects of the formal ZFC universe.

[example: The Universal Class] The formula $x = x$ defines the class $V$ because every set is equal to itself, so for each set $a$ we have $a \in V$ exactly when $a=a$. Suppose, for contradiction, that there were a set $U$ containing every set. Apply the Separation schema to $U$ with the formula $x \notin x$, obtaining the set \begin{align*} R=\{x\in U : x\notin x\}. \end{align*} Since $U$ contains every set and $R$ is a set, we have $R\in U$. By the defining condition for membership in $R$, \begin{align*} R\in R &\iff R\in U \text{ and } R\notin R\\ &\iff R\notin R, \end{align*} because $R\in U$ is true. Thus $R\in R$ holds exactly when $R\notin R$ holds, a contradiction. Hence the definable class $V$ is available in the metatheory, but it cannot be a set in ZFC. [/example]

The example also indicates how the early part of the course will proceed. We will often name large definable collections, such as the class of ordinals or the cumulative hierarchy, while proving separately that particular initial segments are sets.

## Formal Language and Informal Interpretation

The next problem is how to keep formal syntax and mathematical intuition aligned. The formal language of set theory is minimal, but the intended interpretation is rich: membership is the only primitive non-logical relation, and all other notions must be defined from it.

[definition: Language of Set Theory] The language of set theory is the first-order language $\mathcal L_{\in}$ with equality and one binary relation symbol $\in$. [/definition]

Because the language is so small, definitions carry much of the mathematical load. Ordered pairs, functions, relations, natural numbers, ordinals, and cardinals will all be built as sets satisfying formulas in $\mathcal L_{\in}$.

[example: Coding an Ordered Pair] For sets $a$ and $b$, the Kuratowski ordered pair is \begin{align*} (a,b) := \{\{a\}, \{a,b\}\}. \end{align*} [claim]For all sets $a,b,c,d$, one has $(a,b)=(c,d)$ if and only if $a=c$ and $b=d$.[/claim] [proof]If $a=c$ and $b=d$, then \begin{align*} (a,b) &=\{\{a\},\{a,b\}\}\\ &=\{\{c\},\{c,d\}\}\\ &=(c,d). \end{align*} Conversely, suppose \begin{align*} \{\{a\},\{a,b\}\}=\{\{c\},\{c,d\}\}. \end{align*} Since $\{a\}\in\{\{a\},\{a,b\}\}$, the equality gives \begin{align*} \{a\}\in\{\{c\},\{c,d\}\}. \end{align*} Hence either $\{a\}=\{c\}$ or $\{a\}=\{c,d\}$. In the first case, $a=c$. In the second case, $c\in\{c,d\}=\{a\}$, so $c=a$. Thus in all cases, \begin{align*} a=c. \end{align*} Using $a=c$, the assumed equality becomes \begin{align*} \{\{a\},\{a,b\}\}=\{\{a\},\{a,d\}\}. \end{align*} Since $\{a,b\}\in\{\{a\},\{a,b\}\}$, we have \begin{align*} \{a,b\}\in\{\{a\},\{a,d\}\}. \end{align*} Thus either $\{a,b\}=\{a\}$ or $\{a,b\}=\{a,d\}$. If $\{a,b\}=\{a,d\}$, then $b\in\{a,d\}$, so $b=a$ or $b=d$. If $b=d$, we are done. If $b=a$, then \begin{align*} \{a,d\}=\{a,b\}=\{a\}, \end{align*} so $d\in\{a\}$ and therefore $d=a=b$. If $\{a,b\}=\{a\}$, then $b\in\{a\}$, so $b=a$. Also $\{a,d\}\in\{\{a\},\{a,d\}\}=\{\{a\},\{a,b\}\}$, so either $\{a,d\}=\{a\}$ or $\{a,d\}=\{a,b\}$. In the first case, $d=a=b$. In the second case, \begin{align*} \{a,d\}=\{a,b\}=\{a\}, \end{align*} so again $d=a=b$. Therefore $b=d$ in every case. We have shown that $(a,b)=(c,d)$ implies $a=c$ and $b=d$, and the reverse implication was proved above.[/proof] Thus the set $\{\{a\},\{a,b\}\}$ really remembers the first and second coordinates, which is why it can serve as a set-theoretic code for the ordered pair. [/example]

This coding raises a general issue that will recur throughout the course. A definition may choose one convenient representation, but the mathematical content usually lies in the structural properties proved about that representation.