 Large Cardinals Part III of the Mathematical Tripos Lent Term 2023

Lecturer. Benedikt Löwe

Lectures. Monday, Wednesday 11-12.

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 year 2021-22 with the same content; last year's edition has a legady website that includes the handwritten lecture notes. The details will vary between the 2021-22 edition and the 2022-23 edition.

Last year, 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:

1. Paul Minter. Large Cardinals. Lecturer: Benedikt Löwe. Lent term 2022. 2022.
2. Akihiro Kanamori. The Higher Infinite. Large Cardinals in Set Theory from Their Beginnings. Springer 2003 [Springer Monographs in Mathematics].
3. Thomas Jech. Set Theory, The Third Millenium Edition, revised and expanded, Springer 2003 [Springer Monographs in Mathematics].
 L E C T U R E S. Week 1. First Lecture. Monday 23 January 2023. Large cardinal axioms: their goal in light of the incompleteness phenomenon; general idea: generalising largeness properties of $$\omega$$ to uncountable cardinals. Informal description of large cardinal axioms. Two non-examples: sizes of power sets violating $$\mathsf{GCH}$$ (existence not provable in $$\mathsf{ZFC}$$, but not necessarily large) and fixed points of the aleph function (very large, but existence provable in $$\mathsf{ZFC}$$). Reminder: proof that arbitrarily large fixed points of normal ordinal operations exist. Largeness properties of $$\omega$$: limit cardinal, strong limit cardinal, regular. (Remark: smallest aleph fixed point has cofinality $$\aleph_0$$ and is thus singular.) Inaccessible cardinals. Inaccessible cardinals are aleph fixed points. Lecture Notes. Second Lecture. Wednesday 25 January 2023. Using Gödel's Second Incompleteness Theorem to prove that something cannot be proved by $$\mathsf{ZFC}$$. Von Neumann hierarchy and valid axioms of set theory in von Neumann ranks. Second order replacement. If $$\kappa$$ is inaccessible, then $$\mathbf{V}_\kappa$$ satisfies second order replacement. Absoluteness of inaccessibility for von Neumann ranks. Second proof that $$\mathsf{ZFC}$$ does not prove $$\mathsf{IC}$$, not using Gödel's Second Incompleteness Theorem: concrete construction of models of $$\mathsf{ZFC}+\neg\mathsf{IC}$$. Lecture Notes. Week 2. Third Lecture. Monday 30 January 2023. The consistency strength hierarchy: definitions, maximal element, historical connection to Hilbert's finitist program. Consistent theories $$T$$ such that $$T+\mathrm{Cons}(T)$$ is inconsistent. The property of $$\omega$$-consistency (no precise definition). Strictly increasing sequences of length $$\omega$$ under assumption of $$\omega$$-consistency. Strictly increasing sequences of length $$\omega$$ under assumption of $$\mathsf{ZFC}+\mathsf{IC}$$. Definition of worldly cardinals. Lecture Notes. Fourth Lecture. Wednesday 1 February 2023. If $$\mathbf{V}_\kappa\models\mathsf{ZFC}$$, then $$\kappa$$ is a cardinal. In this case, $$\kappa$$ is a limit cardinal and an aleph fixed point (no proof, cf. Example Sheet #1). Basic model-theoretic concepts: elementary equivalence, elementary substructure, elementary embedding. Basic model-theoretic results: Tarski-Vaught Test, elementary chains, Tarski's Chain Lemma. The set of worldly cardinals below an inaccessible cardinal is unbounded. There are worldly cardinals of all cofinalities below $$\kappa$$ among them. Lecture Notes. Week 3. Fifth Lecture. Monday 6 February 2023. Absoluteness, upwards absoluteness, downwards absoluteness. Substructures in model theory. The language of set theory and its lack of function and constant symbols. Defined functions and constant symbols. Non-absoluteness of emptiness for arbitrary models of set theory. Transitive models of set theory. Formula classes: quantifier free, $$\Delta_0$$, $$\Sigma_1$$, $$\Pi_1$$. Semantic formula classes: up to equivalence in a theory $$T$$. Formulas in $$\Delta_0^T$$ are absolute between transitive models of $$T$$. Lecture Notes. Sixth Lecture. Wednesday 8 February 2023. Formulas in $$\Sigma_1^T$$ are upwards absolute between transitive models of $$T$$. Formulas in $$\Pi_1^T$$ are downwards absolute between transitive models of $$T$$. Set theoretic concepts that are in $$\Delta_0$$: function, injection, bijection, cofinal subset; ordinal is in $$\Delta_0^\mathsf{ZFC}$$; set theoretic concepts that are in $$\Pi_0$$: cardinal, regular cardinal, inaccessible cardinal. Transitive models and inner models. Filters, ultrafilters, $$\lambda$$-completeness, principality. Existence of non-principal ultrafilters via Zorn's Lemma (no proof). Measurable cardinals. Measurable cardinals are inaccessible. Lecture Notes. Week 4. Seventh Lecture. Monday 13 February 2023. Infinitary languages: infinitary conjunctions and disjunctions, infinitary quantifiers, syntax and semantics. The expressive power of infinitary languages transcends first-order logic. Weakly compact cardinals. Weakly compact cardinals are inaccessible. The linearity phenomenon in the consistency strength hierarchy. Products and reduced products. Lecture Notes. Eighth Lecture. Wednesday 15 February 2023. Infinitary languages: lengths of $$\mathcal{L}_{\kappa\kappa}$$-formulas; size of $$L_S$$. Łoś's theorem for infinitary languages. Every measurable cardinal is weakly compact. Keisler's Extension Property (KEP). Every weakly compact cardinal has the KEP (without proof). The smallest inaccessible cardinal is not the smallest weakly compact cardinal. Lecture Notes. Week 5. Ninth Lecture. Monday 20 February 2023. Every weakly compact cardinal has the Keisler extension property. Reflection and bootstrapping of reflection. Below every weakly compact cardinal there are unboundedly many inaccessible cardinals. Erdős arrow notation. Relation to Ramsey's theorem. Finite partition cardinals. A cardinal is weakly compact if and only if it is finite partition (without proof). Finite partition cardinals are inaccessible. Lecture Notes. Tenth Lecture. Wednesday 22 February 2023. Closure under diagonal intersections. Normal filters. Measurable cardinals carry a normal ultrafilter (no proof yet). Measurable cardinals with a normal ultrafilter are finite partition. Measurable cardinals and elementary embeddings: $$\kappa<\lambda$$ with $$\kappa$$ measurable and $$\lambda$$ inaccessible is strictly stronger than $$\mathsf{MC}$$. Ultrapower of $$\mathbf{V}_\lambda$$ with $$U$$; basic properties; the ultrapower embedding; wellfoundedness of the ultrapower; transitive ultrapower $$M$$ as Mostowski collapse of the ultrapower; $$M\subseteq\mathbf{V}_\lambda$$. Lecture Notes (note: correction of proof on page 4). Week 6. Eleventh Lecture. Monday 27 February 2023. The ordinals of $$M$$ are $$\lambda$$. The embedding is the identity on $$\mathbf{V}_\kappa$$. The embedding is not the identity: $$j(\kappa)>\kappa$$. Critical point of an embedding. $$\mathbf{V}_{\kappa+1}\subseteq M$$. If $$\lambda$$ is the least inaccessible above $$\kappa$$, then $$\mathbf{V}_\lambda \neq M$$. In $$\mathbf{V}_\lambda$$, the cardinality of $$j(\kappa)$$ is at most $$2^\kappa$$, so $$j(\kappa)$$ is not measurable. Since $$U\notin M$$, we have that $$\mathbf{V}_{\kappa+2}\not\subseteq M$$ (no proof, cf. Example Sheet #3). Reflection arguments: new proof that there are unboundedly many inaccessibles below a measurable. Lecture Notes. Twelfth Lecture. Wednesday 1 March 2023. Is $$\kappa$$ measurable in $$M$$? First approach: meta-argument that "$$\kappa$$ is measurable in $$M$$" cannot be provably. Second approach: surviving cardinals; reflection argument to show that a surviving cardinal has unboundedly many measurable cardinals below. Fundamental Theorem on Measurable Cardinals. Construction of ultrafilter $$U_j$$. Proof that $$U_j$$ is $$\kappa$$-complete non-principal ultrafilter on $$\kappa$$. This filter is always normal (no proof yet). Lecture Notes. Week 7. Thirteenth Lecture. Monday 6 March 2023. The filter derived from an elementary embedding is always normal. Every measurable cardinal carries a normal ultrafilter. Strengthening of the reflection arguments: if $$U$$ is a normal ultrafilter on $$\kappa$$, then the set of inaccessible cardinals below $$\kappa$$ is in $$U$$. Getting rid of the inaccessible cardinal: Scott's trick. Constructing the ultrapower as a transitive class of sets. Definability of the ultrapower and the embedding from $$U$$. Elementarity and the nondefinability of truth. Elementarity as a schema of axioms. The Fundamental Theorem on Measurable Cardinals as a theorem schema. Class theories: von Neumann-Bernays-Gödel ($$\mathsf{NGB}$$) and Kelley-Morse ($$\mathsf{KM}$$) without definitions. Lecture Notes. Thirteenth Lecture. Monday 6 March 2023. The filter derived from an elementary embedding is always normal. Every measurable cardinal carries a normal ultrafilter. Strengthening of the reflection arguments: if $$U$$ is a normal ultrafilter on $$\kappa$$, then the set of inaccessible cardinals below $$\kappa$$ is in $$U$$. Getting rid of the inaccessible cardinal: Scott's trick. Constructing the ultrapower as a transitive class of sets. Definability of the ultrapower and the embedding from $$U$$. Elementarity and the nondefinability of truth. Elementarity as a schema of axioms. The Fundamental Theorem on Measurable Cardinals as a theorem schema. Class theories: von Neumann-Bernays-Gödel ($$\mathsf{NGB}$$) and Kelley-Morse ($$\mathsf{KM}$$) without definitions. Lecture Notes. Week 8. Fifteenth Lecture. Monday 13 March 2023. The survival relation on normal ultrafilters. Mitchell characterisation lemma. The survival relation is a well-founded, irreflexive, transitive relation (without proof). Mitchell order of ultrafilters. Cardinals with higher Mitchell order and their witness objects. Being of positive Mitchell order reflects at a 2-strong cardinal. Strong cardinals. Reinhard cardinals. Kunen's Inconsistency. The Erdős-Hajnal theorem on $$\omega$$-Jónsson functions. Proof of Kunen's Inconsistency. Lecture Notes. Sixteenth Lecture. Wednesday 15 March 2023. Kunen's lemma implies that there cannot be an elementary embedding $$j\colon \mathbf{V}_{\delta+2}\to \mathbf{V}_{\delta+2}$$. Axiom candidates close to the Kunen inconsistency: $$\mathsf{I1}$$ and $$\mathsf{I3}$$. Algebraic nature of these axioms and connection to braid groups. Supercompactness: the supercompact analogue of Reinhardt cardinals cannot exist either. A reflection-based ordering of strength of cardinal properties and its problems: identity crises. The large cardinal hierarchy: an overview. Lecture Notes. Example Sheets & Examples Classes. Examples class #1. Friday 10 February 2023, 1:30-3:30, MR5. Example Sheet #1: pdf file. Examples class #2. Friday 3 March 2023, 1:30-3:30, MR5. Example Sheet #2: pdf file. Examples class #3. Thursday 16 March 2023, 3:30-5:30, MR3. Example Sheet #3: pdf file. Revision class. Thursday 25 May 2023, 1:30-3:30, online via Zoom. Notes: pdf file.