Topics in Set Theory Part III of the Mathematical Tripos Lent Term 2019
Lecturer: Professor Benedikt Löwe, email: bloewe@science.uva.nl
Schedule: Lectures: Monday, Wednesday, Friday, 11–12, MR13.

Example Classes:
#1: Monday 4 February, 3:30–5, MR5. Example Sheet #1
#2: Monday 18 February, 3:30–5, MR5. Example Sheet #2
#3: Monday 4 March, 3:30–5, MR5.
#4: TBD

Revision Session: TBD

 Friday, 18 January 2018 First Lecture. The Continuum Problem and its history. General remark about model constructions in set theory and obstacles created by Gödel's completeness and incompleteness theorems. The language of set theory (no function and constant symbols). Absoluteness, upwards and downwards absoluteness. Quantifier-free formulas and their absoluteness. Remark that even simple formulas such as $$x = \varnothing$$ are not quantifier-free. Second Lecture. For non-transitive models, the formula $$x = \varnothing$$ is not absolute. Transitive models. $$\Delta_0$$-formulas and $$\Delta_0^T$$-formulas. Absoluteness of $$\Delta_0$$-formulas and $$\Delta_0^T$$-formulas for transitive models and transitive models of $$T$$, respectively. Third Lecture. Basic systems of set theory ($$\mathsf{FST}$$, $$\mathsf{Z}$$, $$\mathsf{ZF}$$, $$\mathsf{ZFC}$$). Examples of $$\Delta_0^T$$-formulas: singleton, pair, successor, union, function, injection etc. Wellfoundedness and the definition of ordinals. Under the assumption of the axiom of Foundation, the formula "$$x$$ is an ordinal" is absolute. Fourth Lecture. Absoluteness of wellfoundedness. Absoluteness of operations defined by transfinite recursion from absolute formulas. Defining the $$\models$$ relation inside a transitive model. Absoluteness of the statement $$\mathrm{Cons}(T)$$. Once more Gödel's incompleteness theorem; the theory $$\mathsf{ZFC}^* := \mathsf{ZFC} + \mathrm{Cons}(\mathsf{ZFC})$$ does not prove its own consistency; the theory $$\mathsf{ZFC}+$$"there is a transitive set model of $$\mathsf{ZFC}$$" proves the consistency of $$\mathsf{ZFC}^*$$. Examples of transitive models: the von Neumann hierarchy and collections of hereditarily small sets. Fifth Lecture. If $$\lambda > \omega$$ is a limit ordinal, then $$\mathbf{V}_\lambda\models\mathsf{Z}$$. Failures of Replacement in $$\mathbf{V}_{\omega+\omega}$$ and $$\mathbf{V}_{\omega_1}$$. Strong limit cardinals, inaccessible cardinals. If $$\kappa$$ is inaccessible, then $$\mathbf{V}_\kappa\models\mathsf{ZFC}$$. Sixth Lecture. Getting countable elementary substructures using the Skolem hull and the Tarski-Vaught criterion (Löwenheim-Skolem theorem). Mostowski Collapsing Theorem (without proof). Failure of absoluteness of formulas such as "... is countable" or "... is a cardinal". Seventh Lecture. Observation: the truth value of $$\mathsf{CH}$$ does not change between the surrounding universe and the countable transitive model of $$\mathsf{ZFC}$$ constructed by the Löwenheim-Skolem-Mostowski method from an inaccessible cardinal. Being hereditarily of size $$<\kappa$$, in particular, $$\mathbf{HF}$$ and $$\mathbf{HC}$$. Axioms valid in $$\mathbf{HC}$$: validity of Replacement and failure of Power Set. Eighth Lecture. If $$\kappa$$ is inaccessible, $$\mathbf{H}_\kappa\models\mathsf{ZFC}$$. Tarski's Undefinability of Truth. Undefinability of Definability. Internal definition of definability. First example class. Example Sheet #1 Ninth Lecture. The definable power set $$\mathcal{D}(A)$$ and the definition of the constructible hierarchy. Properties of the constructible hierarchy. Absoluteness of the functions $$\mathrm{Def}$$ and $$\mathcal{D}$$. The axiom of constructibility: characterisation of transitive set models of $$\mathsf{ZFC}+\mathbf{V}{=}\mathbf{L}$$ (part one). Tenth Lecture. Characterisation of transitive models of $$\mathsf{ZFC}+\mathbf{V}{=}\mathbf{L}$$ (part two). Countable transitive models of $$\mathsf{ZFC}+\mathbf{V}{=}\mathbf{L}$$. The axioms of set theory in $$\mathbf{L}_\kappa$$ for $$\kappa$$ inaccessible: Pairing, Union, and Power Set. Eleventh Lecture. The axioms of set theory in $$\mathbf{L}_\kappa$$ for $$\kappa$$ inaccessible: Separation. The Condensation Lemma. Discussion: the Condensation Lemma proved in the meta-set theory proves that $$\mathbf{L}$$ can have at most $$\aleph_1$$ many subsets of the naturals; that is not enough to prove $$\mathsf{CH}$$ in $$\mathbf{L}$$. Twelfth Lecture. Internalisation of the Condensation Lemma and $$\mathsf{GCH}$$ in $$\mathbf{L}$$. Avoiding the use of the inaccessible cardinal: Lévy Reflection Theorem (cf. Example Sheet #2 (15)) and a use of the Compactness Theorem. Proof that the existence of regular limit cardinals cannot be proved in $$\mathsf{ZFC}$$. Thirteenth Lecture. Cancelled due to illness. Thirteenth Lecture. Inner models. Minimality Theorem for $$\mathbf{L}$$. The technique of inner models and its limitations: the technique of inner models cannot show the consistency of $$\neg\mathsf{CH}$$. Illustration: "inner models" and "outer models" of the theory of fields of characteristic zero. Adding an injection from $$\aleph_2^\mathbf{L}$$ to the reals may end up collapsing $$\aleph_1^\mathbf{L}$$. Desiderata: a method of constructing outer models in order to get $$\mathsf{ZFC}$$ and preservation theorems for cardinals Second example class. Example Sheet #2

Last changed: 18 February 2019