Prof Dr Benedikt Loewe: Large Cardinals (L24)

Large Cardinals
Part III of the Mathematical Tripos
Lent Term 2024

Lecturer. Benedikt Löwe

Lectures. Monday, Friday 11-12.

Examples Classes. Ioannis Eleftheriadis.

Description.

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 \(\mathsf{ZFC}\).

Lecture Notes & Literature.

This course was taught in the academical years 2021-22 and 2022-23 with roughly the same content; legacy websites: 2021-22 / 2022-23. Details will vary between the 2021-22 and 2022-23 editions and the 2023-24 edition.

Two years ago, Paul Minter took extensive notes of the course and produced typed lecture notes with his own commentary that he kindly provided for students of the course. The most relevant books are Kanamori's The Higher Infinite and Jech's encyclopedic Set Theory:

Paul Minter. Large Cardinals. Lecturer: Benedikt Löwe. Lent term 2022. 2022.
Akihiro Kanamori. The Higher Infinite. Large Cardinals in Set Theory from Their Beginnings. Springer 2003 [Springer Monographs in Mathematics].
Thomas Jech. Set Theory, The Third Millenium Edition, revised and expanded, Springer 2003 [Springer Monographs in Mathematics].

Crib sheet.

We put together a crib sheet with some results from Part II Logic & Set Theory, Part III Model Theory & Non-Classical Logic, Part III Forcing & the Continuum Hypothesis, and Part III Large Cardinals that will be used in the two Lent term set theory courses. You can find it here: pdf file.

L E C T U R E S.
Week 1.	First Lecture. Friday 19 January 2024. Large cardinal properties and axioms: informal definition using vague terms such as very large and so large that. Interpretation of the vague terms: aleph fixed points, existence of aleph fixed points. Cofinal sets, cofinality, regular and singular cardinals. Weakly and strongly inaccessible cardinals. Every weakly inaccessible cardinal is an aleph fixed point.	Second Lecture. Monday 22 January 2024. The least aleph fixed point has cofinality \(\aleph_0\). Proving consistency by constructing models. The cumulative hierarchy: axioms of set theory in limit von Neumann ranks; failure of replacement in \(\mathbf{V}_{\omega+\omega}\) and \(\mathbf{V}_{\omega_1}\). Second order replacement. Second order replacement implies the scheme of replacement. Zermelo's theorem: if \(\kappa\) is inaccessible, then \(\mathbf{V}_\kappa\) satisfies second order replacement.
Week 2.	Third Lecture. Friday 26 January 2024. Shepherdson's Theorem. Construction of a countable elementary substructure of \(\mathbf{V}_\kappa\): gaps and non-transitivity. Mostowski collapse and countable transitive elementary substructures. Non-absoluteness of "\(x\) is a cardinal". Overview of absoluteness: absoluteness of \(\Delta_0\) formulas for transitive models; downwards absoluteness of \(\Pi_1\) formulas for transitive models. The existence of an inaccessible cardinal is not equivalent to \(\mathrm{Cons}(\mathsf{ZFC})\); in fact, it implies the consistency of \(\mathsf{ZFC}+\mathrm{Cons}(\mathsf{ZFC})\). The countable transitive model cannot be a von Neumann rank.	Fourth Lecture. Monday 29 January 2024. Construction of an \(\alpha\) such that \(\mathbf{V}_\alpha\preccurlyeq\mathbf{V}_\kappa\). Worldly cardinals. For every inaccessible cardinal, there is a worldly cardinal below. Worldly cardinals can have countable cofinality. The consistency strength hierarchy: maximality corresponds to inconsistency, nonmaximal theories \(T\) such that \(T+\mathrm{Cons}(T)\) is inconsistent. \(\mathsf{ZFC} <_\mathrm{Cons} \mathsf{ZFC}+\mathsf{WorC}<_\mathrm{Cons} \mathsf{ZFC}+\mathsf{IC}\).
Week 3.	Fifth Lecture. Friday 2 February 2024. Absoluteness of being inaccessible for von Neumann ranks. If \(\kappa\) is the least inaccessible cardinal, then \(\mathbf{V}_\kappa\models\mathsf{ZFC}+\neg\mathsf{IC}\). The consistency strength of \(\mathsf{ZFC}+\neg\mathsf{IC}\) is strictly weaker than that of \(\mathsf{ZFC}+\mathsf{IC}\). Weakly inaccessible cardinals: under \(\mathsf{GCH}\), every weakly inaccessible is inaccessible. If \(\kappa\) is weakly inaccessible, it is inaccessible in \(\mathbf{L}\). The Lebesgue measure problem. Vitali's negative solution to the measure problem (1905). Remark that Vitali's proof uses \(\mathsf{AC}\). Solovay's result from an inaccessible that \(\mathsf{ZF}+\)"every set is Lebesgue measurable" is consistent; Shelah's result that the inaccessible is necessary for that (no proofs).	Sixth Lecture. Monday 5 February 2024. Banach measures and the Banach measure problem. Every measure is a Banach measure. \(\mathsf{CH}\) implies that the Banach measure problem has a negative solution (Banach-Kuratowski, no proof). \(\lambda\)-additivity. The least cardinal \(\kappa\) with a Banach measure has a \(\kappa\)-additive Banach measure (no proof; cf. Example Sheet #2). Real-valued measurable cardinals. Every real-valued measurable cardinal is regular. Some combinatorial lemmas (among them the existence of Ulam matrices). Every real-valued measurable cardinal is weakly inaccessible. Two-valued Banach measures. Filters, ultrafilters, nonprincipality, \(\lambda\)-completeness. Two-valued Banach measures correspond to nonprincipal ultrafilters.
Week 4.	Seventh Lecture. Friday 9 February 2024. Measurable cardinals. Every measurably cardinal is inaccessible. Diagonal intersections and normality. Every measurable cardinal \(\kappa\) has a normal \(\kappa\)-complete nonprincipal ultrafilter (without proof; cf. Example Sheet #2). Partitions and the Erdös arrow notation. Weakly compact cardinals. Every weakly compact cardinal is inaccessible.	Eighth Lecture. Monday 12 February 2024. Every measurable cardinal is weakly compact. Infinitary languages. Syntax and semantics of infinitary languages. Expressive power of infinitary languages: finiteness can be expressed. Strongly compact cardinals. The Keisler-Tarski theorem: extension of \(\kappa\)-complete filters to \(\kappa\)-complete ultrafilters. Every strongly compact cardinal is measurable.
Week 5.	Ninth Lecture. Friday 16 February 2024. Implication diagram between the current large cardinal notions. Keisler extension property. Reflection: an inaccessible cardinal with the Keisler extension property cannot be the least inaccessible. Reflection argument to show that the set of inaccessibles below is cofinal. Strict inequality in consistency strength: the existence of an inaccessible with the Keisler extension property is strictly stronger than \(\mathsf{IC}\). Every strongly compact cardinal has the Keisler extension property. Weaker compactness property \(\mathsf{WC}(\kappa)\). Weak compactness is equivalent to \(\mathsf{WC}\) (without proof). Consequence: every weakly compact cardinal has the Keisler extension property. \(\mathsf{ZFC}+\mathsf{IC} <_\mathrm{Cons} \mathsf{ZFC}+\mathsf{WC}\). Remark about the existence of unboundedly many worldly cardinals below an inaccessible.	Tenth Lecture. Monday 19 February 2024. Ultrapowers of the universe (in the setting where \(\kappa\) is measurable and \(\lambda>\kappa\) is inaccessible). The ultrapower and its embedding; its wellfoundedness; the transitive set isomorphic to the ultrapower; properties and relationship between the transitive set and \(\mathbf{V}_\lambda\). The embedding is the identity on \(\mathbf{V}_\kappa\).
Week 6.	Eleventh Lecture. Friday 23 February 2024. Critical point. The critical point of \(j\) is \(\kappa\). \(\mathbf{V}_\kappa\in M\). \(\mathbf{V}_{\kappa+1}\subseteq M\). Size of \(j(\kappa)\): at most \(2^\kappa\); i.e., \(j(\kappa)\) is measurable in \(M\), but not in \(\mathbf{V}_\lambda\) and the two models are different. Remark: \(U\notin M\) (no proof).	Twelfth Lecture. Monday 26 February 2024. Reflection at a measurable cardinal. The set of inaccessibles and weakly compacts below a measurable are cofinal. The notion of \(\beta\)-stability. Being weakly compact is 1-stable. Surviving cardinals. Being measurable is 2-stable. The consistency strength of surviving cardinals. Fundamental Theorem on Measurable Cardinals.
Week 7.	Thirteenth Lecture. Friday 1 March 2024. The ultrafilter derived from an embedding is always normal. The operations \(j\mapsto U_j\) and \(U\mapsto j_U\) are in general not inverses of each other. Embedding cardinals. Characterisation of normal ultrafilters via the representation of \(\kappa\). Alternative proofs of reflection using normal ultrafilters. Surviving cardinals. Characterisation of surviving cardinals. Mitchell order on normal ultrafilters. Characterisation of surviving cardinals via Mitchell order. Strong embeddings and strong cardinals. Reflection at strong cardinals.	Fourteenth Lecture. Monday 4 March 2024. Stability, strength, and witness objects. Witness objects for weakly compact, measurable, and surviving cardinals. Getting rid of the inaccessible in the ultrapower construction: Scott's trick to form an ultrapower of a class; Mostowski's collapsing theorem for classes with a set-like, extensional and wellfounded relation. Class theoretic formulations of the Fundamental Theorem on Measurable Cardinals: von Neumann-Bernays-Gödel and Kelley-Morse class theories (without proper definitions).
Week 8.	Fifteenth Lecture. Friday 8 March 2024. Strong cardinals and their witness objects: \(\beta\)-strong cardinals cannot have witness objects in \(\mathbf{V}_\beta\) and strong cardinals cannot have single witness objects. Closure under \(\mu\)-sequences. The ultrapower embedding is closed under \(\kappa\)-sequences, but not under \(\kappa^+\)-sequences. The notions of \(\mu\)-supercompactness and supercompactness. Every \(2^\kappa\)-supercompact cardinal is \(2\)-strong; discussion of the non-local nature of supercompactness. Reinhardt cardinals. Discussion of Jónsson-algebras and \(\omega\)-Jónsson functions. Existence of \(\omega\)-Jónsson functions.	Sixteenth Lecture. Monday 11 March 2024. Kunen's inconsistency. Corollary: there is no elementary embedding from \(\mathbf{V}_{\delta+2}\) to \(\mathbf{V}_{\delta+2}\). Strongest axioms: \(\mathsf{I1}\) and \(\mathsf{I3}\). Connections between \(\mathsf{I3}\) and left-distributive algebras (no proofs). Overview of the large cardinal hierarchy: small large cardinals, medium-size large cardinals, large large cardinals.
Example Sheets & Examples Classes.	Example Sheet #1. pdf file. Solutions provided by Ioannis Eleftheriadis: pdf file. Example Sheet #2. pdf file. Solutions provided by Ioannis Eleftheriadis: pdf file. Example Sheet #3. pdf file. Solutions provided by Ioannis Eleftheriadis: pdf file.