Large Cardinals
Part III of the Mathematical Tripos
Lent Term 2022

Lecturer. Benedikt Löwe

Lectures. Monday, Wednesday 11-12.

Examples Classes. Ioannis Eleftheriadis.

The following definitions and facts should be familiar from an introductory course on set theory: a cardinal \(\kappa\) is called regular if every unbounded subset of \(\kappa\) has cardinality \(\kappa\); successor cardinals, i.e., cardinal s of the form \(\aleph_{\alpha+1}\), are always regular; the usual limit cardinals, e.g., \(\aleph_\omega\), \(\aleph_{\omega+ \omega}\), or \(\aleph_{\omega_1}\), are not.

Thus, the following is a natural question: "Are there any uncountable regular limit cardinals?''. If they exist, they must be very large, in particular, much larger than any of the mentioned limit cardinals. It turns out that this question is intricately connected with the incompleteness phenomenon in set theory: if there is an uncountable regular limit cardinal, then there is a model of \(\mathsf{ZFC}\); therefore, \(\mathsf{ZFC}\) is consistent, and hence (by Gödel's Second Incompleteness Theorem) \(\mathsf{ZFC}\) cannot prove the existence of these cardinals (unless, of course, it is inconsistent). Regular limit cardinals (a.k.a. weakly inaccessible cardinals) are the smallest examples of set-theoretic notions called large cardinals: cardinals with properties that imply that they must be very big and whose existence cannot be proved in \(\mathsf{ZFC}\). In this course, we shall get to know a number of these large cardinals, study their behaviour, observe consequences of their existence for set theory, and develop techniques to determine the logical strength of large cardinals (the so-called consistency strength hierarchy). In modern set theory, large cardinals are used as the standard way to calibrate logical strength of extensions of \(\mathsf{ZFC}\).

Prerequisites. The Part II course Logic & Set Theory or an equivalent course is essential. The Part III course Computability & Logic lectured in Lent 2022 is not directly related to the material of this course, but provides a useful supplementary glance at the incompleteness phenomenon that underlies the consistency strength hierarchy, so students would benefit from taking these two courses in tandem.


  1. Thomas Jech, Set Theory, The Third Millenium Edition, revised and expanded, Springer 2003 [Springer Monographs in Mathematics].
  2. Akihiro Kanamori, The Higher Infinite. Large Cardinals in Set Theory from Their Beginnings. Springer 2003 [Springer Monographs in Mathematics]

Teaching mode. The teaching mode of this course will be synchronous online teaching with additional in-person office hours: all lectures will be offered synchronously via Zoom. Room MR 14 is reserved for the course during the regular lecture hours (Monday & Wednesday 11–12) and students can sit there and watch the lecture on their computers with head-phones on. This moodle page will contain the Zoom link for the lectures (in the section Lectures). Lectures will also be recorded and posted on Panopto within 24 hours of the lecture.

L E C T U R E S.
Week 1. Lecture I. Informal definition of "large cardinals". Normal ordinal operations and aleph fixed points. Aleph fixed points are very big, but not too big for ZFC. Cofinal sets, cofinality, regular and singular cardinals. Weakly inaccessible cardinals. Weakly inaccessible cardinals are aleph fixed points. Strongly inaccessible cardinals. Lecture Notes.
Teaching mode: Online only.
Monday 24 January 2022. 11–12.
Lecture II. Von Neumann hierarchy, its properties, and von Neumann levels as models of set theory. Hausdorff's Theorem: if \(\kappa\) is inaccessible, \(\mathbf{V}_\kappa\) is a model of ZFC. Two proofs that IC cannot be proved in ZFC: one using Gödel's Second Incompleteness Theorem and one using absoluteness of inaccessibility. Worldly cardinals. Each inaccessible cardinal has many worldly cardinals below it (no proof yet). Lecture Notes.
Teaching mode: Online only.
Wednesday 26 January 2022. 11–12.
Week 2. Lecture III. Worldliness implies that \(\kappa\) is a cardinal. Worldliness implies that \(\kappa\) is a limit cardinal (without proof; cf. Example Sheet #1). Model theory: elementary equivalence and elementary substructures; Tarski-Vaught Test (proof sketch). Construction of a von Neumann level that elementary substructure. The least worldy is not the least inaccessible. The cofinality of the worldly cardinal constructed like this is \(\aleph_0\). Discussion of regular worldly cardinals. Lecture Notes.
Teaching mode: Online Only.
Monday 31 January 2022. 11–12.
Lecture IV. Some basic model theory: Tarski's Chain Lemma (without proof). Construction of worldly cardinals with uncountable cofinality. Transitive submodels. Closure of families of formulas. \(\Delta_0\) formulas. \(\Delta_0\) formulas are absolute between transitive submodels. Being an ordinal is \(\Delta_0\). Lecture Notes.
Teaching mode: Online Only.
Wednesday 2 February 2022. 11–12.

Office Hour.
Teaching mode: In person in MR14.
Friday 4 February 2022. 11–12.

Week 3. Lecture V.
Teaching mode: Online Only.
Monday 7 February 2022. 11–12. Non-preservation of simple formulas for non-transitive sets. The generalised continuum hypothesis, strong limits, and inaccessible cardinals. \(\Pi_1\) and \(\Sigma_1\) formulas. Downwards and upwards absoluteness. \(\Pi_1\) concepts: cardinal, regular, limit, strong limit. Inner models and definable models. Gödel's constructible universe and the consistency of the generalised continuum hypothesis (no proof). The existence of weakly inaccessible cardinals cannot be proved in \(\mathsf{ZFC}\). Improved Löwenheim-Skolem Theorem: countable elementary substructures via the Tarski-Vaught Test. Construction of a non-transitive submodel of \(\mathsf{ZFC}\). Lecture Notes.
Lecture VI.
Teaching mode: Online Only.
Wednesday 9 February 2022. 11–12.The measure problem: original question and Vitali's negative answer assuming \(\mathsf{AC}\) (no proof). Banach's abstract version; Ulam measures; Ulam cardinals. \(\kappa\)-additivity; measurable cardinals. Measurable cardinals are inaccessible. Compactness: syntax of \(\mathcal{L}_{\kappa\lambda}\) languages. Lecture Notes.
Week 4. Lecture VII.
Teaching mode: Online Only.
Monday 14 February 2022. 11–12. Compactness: Semantics of \(\mathcal{L}_{\kappa\lambda}\) languages. Strong compactness and weak compactness. Weak compactness implies regularity. The Keisler-Tarski Theorem. Strong compactness implies measurability. Lecture Notes.
Lecture VIII.
Teaching mode: Online Only.
Wednesday 16 February 2022. 11–12. Weakly compact cardinals are inaccessible. Ultraproducts and ultrapowers: Łoś's Theorem (no proof), elementary embedding from a structure into its ultrapower. Measurable cardinals are inaccessible. Lecture Notes.

Office Hour.
Teaching mode: In person in MR14.
Friday 18 February 2022. 11–12.

Week 5. Lecture IX.
Teaching mode: Online Only.
Monday 21 February 2022. 11–12. Keisler Extension Property. Every weakly compact has the Keisler Extension Property. The smallest inaccessible is not weakly compact. Hierarchies of logical strength: consequence strength and its non-linearity; strength by size of the smallest cardinal. Lecture Notes.
Lecture X.
Teaching mode: Online Only.
Wednesday 23 February 2022. 11–12. Silly examples that show problems with the hierarchy defined by smallest size; identity crises. Consistency strength hierarchy; non-linearity of the consistency strength hierarchy (no proof). Measurable cardinals and elementary embeddings: identifying the ultrapower as a subset of \(\mathbf{V}_\lambda\); extensionality in the ultrapower. Lecture Notes.
Week 6. Lecture XI.
Teaching mode: Online Only.
Monday 28 February 2022. 11–12. Wellfoundedness of the ultrapower. The transitive model \(M\), the ultrapower embedding \(j_U\), and their properties: size of \(M\), size of elements of \(M\), height of \(M\), identity of \(j_U\) on \(\mathbf{V}_\kappa\). Nontrivial embeddings and critical points: \(\kappa\) is the critical point of \(j_U\). Fundamental Theorem on Measurable Cardinals. Lecture Notes.
Lecture XII.
Teaching mode: Online Only.
Wednesday 2 March 2022. 11–12. Fundamental Theorem on Measurable Cardinals (embedding implies existence of ultrafilter). Properties of the ultrapower: \(j(\kappa)\) is measurable in \(M\), \(\kappa\) is inaccessible in \(M\), \(\mathbf{V}_{\kappa+1}\subseteq M\), \(\kappa^+\) is the successor of \(\kappa\) in \(M\), the cardinality of \(j(\kappa)\) is at most \(2^\kappa\), \(\mathbf{V}_\lambda\neq M\). Lecture Notes.

Office Hour.
Teaching mode: In person in MR14.
Friday 4 March 2022. 11–12.

Week 7. Lecture XIII.
Teaching mode: Online Only.
Monday 7 March 2022. 11–12. The ultrafilter is not in the ultrapower; \(\mathbf{V}_{\kappa+2}\not\subseteq M\). Can \(\kappa\) still be measurable? Surviving cardinals: a reflection argument shows that surviving cardinals are the \(\kappa\)th measurable cardinal. Preservation of weak compactness: if weak compactness is preserved, then reflection shows that measurables are the \(\kappa\)th weakly compact. Lecture Notes.
Lecture XIV.
Teaching mode: Online Only.
Wednesday 9 March 2022. 11–12. The behaviour of \(\mathcal{L}_{\kappa\kappa}\) languages and their sets of formulas under elementary embeddings: the image of the satisfying model witnesses satisfiability in \(M\). Strengthenings of measurability. Limits of measurables: \(\alpha\)-measurability; surviving cardinals \(\kappa\) are \(\kappa\)-measurable. Mitchell order: survival of ultrafilters on \(\kappa\) is absolute for transitive models containing \(\mathbf{V}_{\kappa+1}\); higher Mitchell order is of strictly higher strength. Strength: \(\alpha\)-strong embeddings; \(\alpha\)-strong cardinals; measurability is equivalent to 1-strength. Lecture Notes.
Week 8. Lecture XV.
Teaching mode: Online Only.
Monday 14 March 2022. 11–12. If \(\kappa\) is 2-strong, it is \(\kappa\)-measurable. Relationship between Mitchell order and strength (without proof). Strong cardinals: different levels of strength witnessed by different embeddings. Reinhardt cardinals. Kunen's inconsistency (proof modulo existence of \(\omega\)-Jónsson functions). Bound on the strength of a given embedding. Supercompactness; supercompactness implies strength. Measurable cardinals \(\kappa\) are \(\kappa\)-supercompact. Lecture Notes.
Lecture XVI.
Teaching mode: Online Only.
Wednesday 16 March 2022. 11–12. Large cardinals and \(\mathsf{GCH}\): Scott's Theorem (measurables cannot be the least counterexample to \(\mathsf{GCH}\); strengthening of Scott's Theorem for supercompact cardinals. Witness objects: discussion of role of the Fundamental Theorem for the method of reflection; witness objects for supercompactness (without proper definition and proof), strong compactness (without proper definition and proof), strength (without proper definition and theorem statement). The Fundamental Theorem without the inaccessible: proper classes, definable proper classes, reformulation of the Fundamental Theorem as an infinite list of meta-statements. Taking the ultrapower of the universe: Scott's trick. Lecture Notes.
Example Sheets & Examples Classes. Examples Class #1
Example Sheet #1: pdf file.
Solutions: pdf file.
Friday 11 February 2022, MR 4, 15:30–17:00

Examples Class #2
Example Sheet #2: pdf file.
Solutions: pdf file.
Friday 25 February 2022, MR 4 online via Zoom, 15:30–17:00

Examples Class #3
Example Sheet #3: pdf file.
Solutions: pdf file.
Friday 18 March 2022, MR 4, 15:30–17:00

Revision Class
Friday 20 May 2022, MR 9, 15:30–17:00