Lectures on Set Theory, Beijing, June 2025

Posted on June 11, 2025 by Joel David Hamkins

This will be a lecture series at Peking University in Beijing in June 2023.

Course abstract. This will be a series of advanced lectures on set theory, treating diverse topics and particularly those illustrating how set theoretic ideas and conceptions shed light on core foundational matters in mathematics. We will study the pervasive independence phenomenon over the Zermelo-Fraenkel axioms of set theory, perhaps the central discovery of 20th century set theory, as revealed by the method of forcing, which we shall study in technical detail with numerous examples and applications, including iterated forcing. We shall look into all matters of the continuum hypothesis and the axiom of choice. We shall introduce the basic large cardinal axioms, those strong axioms of infinity, and investigate the interaction of forcing and large cardinals. We shall explore issues of definability and truth, revealing a surprisingly malleable nature by the method of forcing. Looking upward from a model of set theory to all its forcing extensions, we shall explore the generic multiverse of set theory, by which one views all the models of set theory as so many possible mathematical worlds, while seeking to establish exactly the modal validities of this conception. Looking downward in contrast transforms this subject to set-theoretic geology, by which one understands how a given set-theoretic universe might have arisen from its deeper grounds by forcing. We shall prove the ground-model definability theorem and the other fundamental results of set-theoretic geology. The lectures will assume for those participating a certain degree of familiarity with set-theoretic notions, including the basics of ZFC and forcing.

There will be ten lectures, each a generous 3 hours.

Lecture 1. Set Theory

This first lecture begins with fundamental notions, including the dramatic historical developments of set theory with Cantor, Frege, Russell, and Zermelo, and then the rise of the cumulative hierarchy and the iterative conception. The move to a first-order foundational theory. The Skolem paradox. The omission of urelements and the move to a pure set theory. We will establish the reflection phenomenon and the phenomenon of correctness cardinals, before providing some simple relative consistency results. We will compare the first-order approach to the various class theories and also lay out the spectrum of weak theories, including locally verifiable set theory, before discussing countabilism as an approach to set theory.

Lecture 2. Categoricity and the small large cardinals

We will discuss the central role and importance of categoricity in mathematics, highlighting this with results of Dedekind and Huntington, and with several examples of internal categorcity. Afterwards, we shall begin to introduce various small large cardinal notions—the inaccessible cardinals, the hyperinaccessibility hierarchy, Mahlo cardinals, worldly cardinals, other-worldly cardinals. We shall explain the connection with categoricity via Zermelo’s categoricity result. Going deeper, we discuss the possibility of categorical large cardinals and the enticing possibility of a fully categorical set theory.

Lecture 3. Forcing

We shall give an introduction to forcing, pursuing and comparing two approaches, via partial orders versus Boolean algebras. Forcing arises naturally from the iterative conception of the cumulative hierarchy, when undertaken in multi-valued logic. We shall see the principal introductory forcing examples, including the forcing to add a Cohen real, cardinal collapse forcing, forcing the failure of CH, forcing to add dominating reals, almost disjoint coding, iterated forcing, the forcing of Martin’s axiom, and the case of Suslin trees.

Lecture 4. Continuum Hypothesis

We tell the story of the continuum hypothesis, from Cantor’s initial conception and strategy, to Gödel’s proof of CH in the constructible universe, and ultimately Cohen’s forcing of ¬CH, establishing independence over ZFC. The CH is a forcing switch. We discuss the generalized continuum hypothesis GCH, and prove Easton’s theorem on the continuum function. Finally, we discuss various philosophical approaches to settling the CH problem, including Freiling’s axiom and the equivalence with ¬CH, and the role of the continuum hypothesis in providing a categorical theory of the hyperreals. Two equivalent formulations of CH in ZFC are not equivalent without AC.

Lecture 5. Axiom of Choice

We tell the story of the axiom of choice, beginning with a spectrum of equivalent formulations, including the linearity of cardinality. We discuss the abstract cardinal-assignment problem versus the cardinal-selection problem. We establish the truth of the axiom of choice in the constructible universe, as well as global choice, but ultimately the independence of the axiom of choice over ZF via forcing and the symmetric model construction method. Finally, we discuss the perfect predictor theorem and the box puzzle conundrum.

Lecture 6. Definability

We shall define and discuss the formal notion of definability in mathematics and set theory. Can every set be definable? We exhibit the phenomenon of pointwise definable models and their relevance for the Math Tea argument. We define the inner model HOD and explore its interaction with forcing, forcing V=HOD and also forcing V≠HOD. We reveal the coquettish nature of HOD, establishing the nonabsoluteness of HOD, showing furthermore that every model of set theory is the HOD of another model. We show how forcing generic filters can be definable in their forcing extensions. Finally, we shall exhibit a spectrum of paradoxical examples revealing various subtleties in the notion of definability.

Lecture 7. Truth

What is truth? We establish Tarski’s theorem on the nondefinability of truth, and establish the second incompleteness theorem via the Grelling-Nelson paradox. We analyze the connection between truth predicates and correctness cardinals. What is the consistency strength of having a truth predicate? Can a model of set theory contain its own theory as an element? Must it? We define the truth telling game. We shall force a definable truth predicate for HOD. We shall establish the nonabsoluteness of satisfaction.

Lecture 8. Forcing and large cardinals

Can large cardinals settle CH? Gödel had hoped so, but this is refuted by the Levy-Solovay theorem. We will prove forcing preservation theorems for large cardinals, and nonabsoluteness theorems. On the difference between lifting and extending measures. Laver indestructibility and the lottery preparation, via master condition arguments.

Lecture 9. Set-theoretic geology

Looking down, we shall give an introduction to set-theoretic geology. We will prove the ground model definability theorem, using the cover and approximation properties. We shall define the Mantle and prove that every model of set theory is the Mantle of another model. We will discuss Bukovski’s theorem characterizing forcing extensions and prove Usuba’s theorems on the downward directedness of grounds.

Lecture 10. Set-theoretic potentialism

Looking up, we view forcing as a modality, viewing every model of set theory in the context of its generic multiverse. We shall investigate the modal logic of forcing with independent buttons and switches. We shall explore the other natural interpretations of set-theoretic potentialism and investigate their modal validities.

Comments or suggestions welcome.

Infinity, University of Notre Dame, Spring 2023

Posted on January 8, 2023 by Joel David Hamkins

Infinity

Philosophy 20607 01 (32582)

University of Notre Dame Spring 2023

Instructor: Joel David Hamkins, O’Hara Professor of Philosophy and Mathematics
3:30-4:45 Tuesdays + Thursdays, DeBartolo Hall 208

Course Description. This course will be a mathematical and philosophical exploration of infinity, covering a wide selection of topics illustrating this rich, fascinating concept—the mathematics and philosophy of the infinite.

Along the way, we shall find paradox and fun—and all my favorite elementary logic conundrums and puzzles. It will be part of my intention to reveal what I can of the quirky side of mathematics and logic in its connection with infinity, but with a keen eye open for when issues happen to engage with philosophically deeper foundational matters.

The lectures will be based on the chapters of my forthcoming book, The Book of Infinity, currently in preparation, and currently being serialized and made available on the Substack website as I explain below.

Topics. Among the topics we shall aim to discuss will be:

The Book of Numbers
Zeno’s paradox
The infinite coastline paradox
Supertasks
Largest number contest
The googol plex chitty bang stack hierarchy
Galileo’s Salviati on infinity
Hilbert’s Grand Hotel
The uncountable
How to count (to infinity and beyond!)
Slaying the Hydra
Transfinite recursion
The continuum hypothesis
The axiom of choice
Orders of infinity
The lattice of subsets of ℕ
Potential versus actual infinity
Confounding puzzles of infinity
Infinite liars
Infinite utilitarianism
Infinite computation
Infinite games
Indescribable numbers
Extremely remote events of enormous consequence
The sand reckoner
Paradox in high dimension
The outer limits of reason
Puzzles of epistemic logic and the problem of common knowledge

Mathematical background. The course will at times involve topics and concepts of a fundamentally mathematical nature, but no particular mathematical background or training will be assumed. Nevertheless, it is expected that students be open to mathematical thinking and ideas, and furthermore it is a core aim of the course to help develop the student’s mastery over various mathematical concepts connected with infinity.

Readings. The lectures will be based on readings from the topic list above that will be made available on my Substack web page, Infinitely More. Readings for the topic list above will be gradually released there during the semester. Each reading will consist of a chapter essay my book-in-progress, The Book of Infinity, which is being serialized on the Substack site specifically for this course. In some weeks, there will be supplemental readings from other sources.

Student access. I will issue subscription invitations to the Substack site for all registered ND students using their ND email, with free access to the site during the semester, so that students can freely access the readings. Students are free to manage their subscriptions however they see fit. Please inform me of any access issues. There are some excellent free Substack apps available for Apple iOS and Android for reading Substack content on a phone or other device.

Discussion forum. Students are welcome to participate in the discussion forums provided with the readings to discuss the topics, the questions, to post answer ideas, or engage in the discussion there. I shall try to participate myself by posting comments or hints.

Homework essays. Students are expected to engage fully with every topic covered in the class. Every chapter concludes with several Questions for Further Thought, with which the students should engage. It will be expected that students complete approximately half of the Questions for Further thought. Each question that is answered should be answered essay-style with a mini-essay of about half a page or more.

Extended essays. A student may choose at any time to answer one of the Questions for Further Thought more fully with a more extended essay of two or three pages, and in this case, other questions on that particular topic need not be engaged. Every student should plan to exercise this option at least twice during the semester.

Final exam. There will be a final exam consisting of questions similar to those in the Questions for Further Thought, covering every topic that was covered in the course. The final grade will be based on the final exam and on the submitted homework solutions.

Open Invitation. Students outside of Notre Dame are welcome to follow along with the Infinity course, readings, and online discussion. Simply subscribe at Infinitely More, keep up with the readings and participate in the discussions we shall be having in the forums there.

The continuum hypothesis and other set-theoretic ideas for non-set-theorists, CUNY Einstein Chair Seminar, April, 2015

Posted on April 21, 2015 by Joel David Hamkins

At Dennis Sullivan’s request, I shall speak on set-theoretic topics, particularly the continuum hypothesis, for the Einstein Chair Mathematics Seminar at the CUNY Graduate Center, April 27, 2015, in two parts:

An introductory background talk at 11 am, Room GC 6417
The main talk at 2 – 4 pm, Room GC 6417

I look forward to what I hope will be an interesting and fruitful interaction. There will be coffee/tea and lunch between the two parts.

Abstract. I shall present several set-theoretic ideas for a non-set-theoretic mathematical audience, focusing particularly on the continuum hypothesis and related issues.

At the introductory background talk, in the morning (11 am), I shall discuss and prove the Cantor-Bendixson theorem, which asserts that every closed set of reals is the union of a countable set and a perfect set (a closed set with no isolated points), and explain how it led to Cantor’s development of the ordinal numbers and how it establishes that the continuum hypothesis holds for closed sets of reals. We’ll see that there are closed sets of arbitrarily large countable Cantor-Bendixson rank. We’ll talk about the ordinals, about $ω_{1}$ , the long line, and, time permitting, we’ll discuss Suslin’s hypothesis.

At the main talk, in the afternoon (2 pm), I’ll begin with a discussion of the continuum hypothesis, including an explanation of the history and logical status of this axiom with respect to the other axioms of set theory, and establish the connection between the continuum hypothesis and Freiling’s axiom of symmetry. I’ll explain the axiom of determinacy and some of its applications and its rich logical situation, connected with large cardinals. I’ll briefly mention the themes and goals of the subjects of cardinal characteristics of the continuum and of Borel equivalence relation theory. If time permits, I’d like to explain some fun geometric decompositions of space that proceed in a transfinite recursion using the axiom of choice, mentioning the open questions concerning whether there can be such decompositions that are Borel.

Dennis has requested that at some point the discussion turn to the role of set theory in the foundation for mathematics, compared for example to that of category theory, and I would look forward to that. I would be prepared also to discuss the Feferman theory in comparison to Grothendieck’s axiom of universes, and other issues relating set theory to category theory.

The global choice principle in Gödel-Bernays set theory

Posted on December 3, 2014 by Joel David Hamkins

I’d like to follow up on several posts I made recently on MathOverflow (see here, here and here), which engaged several questions of Gérard Lang that I found interesting. Specifically, I’d like to discuss a number of equivalent formulations of the global choice principle in Gödel-Bernays set theory. Let us adopt the following abbreviations for the usually considered theories:

GB is the usual Gödel-Bernays set theory without any choice principle.
GB+AC is GB plus the axiom of choice for sets.
GBC is GB plus the global choice principle.

The global choice principle has a number of equivalent characterizations, as proved in the theorem below, but for definiteness let us take it as the assertion that there is a global choice function, that is, a class $F$ which is a function such that $F (x) \in x$ for every nonempty set $x$ .

Note in particular that I do not use the set version of choice AC in the equivalences, since most of the statements imply AC for sets outright (except in the case of statement 7, where it is stated specifically in order to make the equivalence).

Theorem. The following are equivalent over GB.

The global choice principle. That is, there is a class function $F$ such that $F (x) \in x$ for every nonempty set $x$ .
There is a bijection between $V$ and $Ord$ .
There is a global well-ordering of $V$ . That is, there is a class relation $◃$ on $V$ that is a linear order, such that every nonempty set has a $◃$ -least member.
There is a global set-like well-ordering of $V$ . There is a class well-ordering $◃$ as above, such that all $◃$ -initial segments are sets.
Every proper class is bijective with $Ord$ .
Every class injects into $Ord$ .
AC holds for sets and $Ord$ injects into every proper class.
$Ord$ surjects onto every class.
Every class is comparable with $Ord$ by injectivity; that is, one injects into the other.
Any two classes are comparable by injectivity.

Proof.

( $1 \to 2$ ) Assume that $F$ is a global choice class function. Using the axiom of replacement, we may recursively define a class sequence of sets $⟨ x_{α} ∣ α \in Ord ⟩$ , where $x_{α} = F (X_{α})$ , where $X_{α}$ is the set of minimal-rank sets $x$ not among ${x_{β} ∣ β < α}$ . That is, we use $F$ to choose the next element among the minimal-rank sets not yet chosen. Thus, we have an injection of $V$ into $Ord$ . If a set $x$ does not appear as some $x_{α}$ on this sequence, then no set of that rank or higher can appear, since we always add sets of the minimal rank not yet having appeared; thus, in this case we will have injected $Ord$ into some $V_{β}$ . But this is impossible by Hartog’s theorem, and so in fact we have bijection between $Ord$ and $V$ .

( $2 \to 3$ ) If there is a bijection between $Ord$ and $V$ , then we may define a global well-ordering by $x < y$ if $x$ appears before $y$ in that enumeration.

( $3 \to 1$ ) Let $F (x)$ be the least element of $x$ with respect to a fixed global well-ordering.

( $3 \to 4$ ) If there is a global well-ordering $<$ , then we may refine it to a set-like well-ordering, by defining $x ◃ y$ just in case the rank of $x$ is less than the rank of $y$ , or they have the same rank and $x < y$ . This relation is still a well-order, since the least member of any nonempty set $X$ will be the $◃$ -least member of the set of members of $X$ having minimal rank. The relation $◃$ is set-like, because the $◃$ -predecessors of any set $x$ are amongst the sets having rank no larger than $x$ , and this is a set.

( $4 \to 5$ ) If there is a global set-like well-ordering $<$ of $V$ and $X$ is a proper class, then $<$ on $X$ is a well-ordering of $X$ , and we may map any ordinal $α$ to the $α^{t h}$ member of $X$ . This will be a bijection of $Ord$ with $X$ .

( $5 \to 6$ ) If every proper class is bijective with $Ord$ , then $V$ is bijective with $Ord$ , and so every set injects into $Ord$ by restriction.

( $6 \to 7$ ) If every class injects into $Ord$ , then in particular, $V$ injects into $Ord$ . The image of this injection is a proper class subclass of $Ord$ , and all such classes are bijective with $Ord$ by mapping $α$ to the $α^{t h}$ member of the class, and so every proper class is bijective with $Ord$ . So $Ord$ injects the other way, and also AC holds.

( $7 \to 3$ ) Suppose that AC holds and $Ord$ injects into every proper class. Let $W$ be the class of all well-orderings of some rank-initial segment $V_{α}$ of the set-theoretic universe $V$ . Since for each $α$ there are only a set number of such well-orderings of $V_{α}$ , if we inject $Ord$ into $W$ , then there must be well-orderings of unboundedly many $V_{α}$ in the range of the injection. From this, we may easily construct a global well-ordering of $V$ , by defining $x < y$ just in case $x$ has lower rank than $y$ , or they have the same rank and $x < y$ in the first well-ordering of a sufficiently large $V_{α}$ to appear in the range of the injection.

( $5 \to 8$ ) Immediate.

( $8 \to 3$ ) If $Ord$ surjects onto $V$ , then there is a global well-ordering, defined by $x < y$ if the earliest appearance of $x$ in the surjection is earlier than that of $y$ .

( $6 \to 9$ ) Immediate.

( $9 \to 3$ ) Assume every class is comparable with $Ord$ via injectivity. It follows that AC holds for sets, since $Ord$ cannot inject into a set, and if a set injects into $Ord$ then it is well-orderable. Now, if $Ord$ injects into the class $W$ used above, consisting of all well-orderings of a $V_{α}$ , then we saw before that we can build a well-ordering of $V$ . And if $W$ injects into $Ord$ , then $W$ is well-orderable and we can also in this case build a well-ordering of $V$ .

( $5 \to 10$ ) If $V$ is bijective with $Ord$ , then every class is bijective with $Ord$ or with an ordinal, and these are comparable by injections. So any two classes are comparable by injections.

( $10 \to 9$ ) Immediate.

QED

Let’s notice a few things.

First, we cannot omit the AC assertion in statement 7. To see this, consider the model $V = L (R)$ , in a case where it does not satisfy AC. I claim that in this model, $Ord$ injects into every proper class that is definable from parameters. The reason is that every object in $L (R)$ is definable from ordinal and real parameters, and indeed, definable in some $V_{α}^{L (R)}$ by some real and ordinal parameters. Indeed, one needs only one ordinal and real parameter. If $W$ is any proper class, then there
must be a proper subclass $W_{0} \subset W$ whose elements are all defined by the same definition in this way. And by partitioning further, we may find a single real that works with various ordinal parameters using that definition to define a proper class of
elements of $W$ . Thus, we may inject $Ord$ into $W$ , even though AC fails in $L (R)$ .

Second, the surjectivity analogues of a few of the statements are not equivalent to global choice. Indeed, ZF proves that every proper class surjects onto $Ord$ , with no choice at all, since if $W$ is a proper class, then there are unboundedly many ordinals
arising as the rank of an element of $W$ , and so we may map each element $x \in W$ to $α$ , if the rank of $x$ is the $α^{t h}$ ordinal that is the rank of any element of $W$ .

The rigid relation principle, a new weak choice principle

Posted on September 25, 2011 by Joel David Hamkins

[bibtex key=HamkinsPalumbo2012:TheRigidRelationPrincipleANewWeakACPrinciple]

The rigid relation principle, introduced in this article, asserts that every set admits a rigid binary relation. This follows from the axiom of choice, because well-orders are rigid, but we prove that it is neither equivalent to the axiom of choice nor provable in Zermelo-Fraenkel set theory without the axiom of choice. Thus, it is a new weak choice principle. Nevertheless, the restriction of the principle to sets of reals (among other general instances) is provable without the axiom of choice.

This paper arose out of my related mathoverflow question: Does every set admit a rigid binary relation (and how is this related to the axiom of choice)?

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Infinity

Philosophy 20607 01 (32582)

University of Notre Dame Spring 2023

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