 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. Example Sheet #3
#4: Thursday 14 March, 3:30–5, MR20. Example Sheet #4

Revision Session: TBD

 Friday, 18 January 2019 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 Fourteenth Lecture. Forcing partial orders, antichains, dense sets, filters, generic sets, splitting partial orders. Examples: Cohen forcing, the forcing collapsing a set $$X$$ to be countable. Generic filters over splitting forcing partial orders cannot lie in the ground model. Fifteenth Lecture. Generic filters for countably many dense sets (or over countable transitive models) exist. The class of names. Simplest examples: the one-element partial order and the Boolean algebra with four elements. The valuation of names; example: names for subsets of $$\{\varnothing\}$$. Canonical names. Sixteenth Lecture. Name for the generic filter. Transitivity of $$M[G]$$. Height of $$M[G]$$. Pairing and Union in $$M[G]$$. The (semantic) forcing relation. The Forcing Theorem (statement). Seventeenth Lecture. Basic properties of the semantic forcing relation. Notion of dense below $$p$$. Definition of the syntactic forcing relation. Eighteenth Lecture. Basic properties of the syntactic forcing relation. The forcing theorem implies that syntactic and semantic forcing relation are equivalent. Proof of the forcing theorem: Cases $$\wedge$$, $$\neg$$, $$\exists$$ (first half). Nineteenth Lecture. Proof of the forcing theorem: Cases $$\exists$$ (second half), $$\in$$, $$=$$. Third example class. Example Sheet #3 Twentieth Lecture. End of the proof of the forcing theorem. The generic model theorem and its consequences for minimality of the forcing extension. Proof of the generic model theorem: some of the axioms were proved earlier on on Example Sheets #3 and #4, proof of Infinity, Separation, and Power Set. Twenty-first Lecture. Discussion of the forcing collapsing $$\aleph_1^M$$. The forcing adding $$\aleph_2^M$$ many new reals. Preservation of cardinals. Preservation theorem for c.c.c. forcings (proof from main lemma). Statement of the main lemma on c.c.c. forcings (no proof). Twenty-second Lecture. Proof of main lemma on c.c.c. forcings. Questions left open by the proof of the consistency of $$\neg\mathsf{CH}$$. Nice names and construction of a model of $$2^{\aleph_0} = \aleph_2$$. Twenty-third Lecture. Construction of models of $$2^{\aleph_0} = \aleph_n$$. Kőnig's Lemma and its special case $${\aleph_\omega}^{\aleph_0} > \aleph_\omega$$; as a consequence $$2^{\aleph_0} \neq \aleph_\omega$$. Adding $$\aleph_\omega$$ many reals to a model of $$\mathsf{GCH}$$ gives $$2^{\aleph_0} = \aleph_{\omega+1}$$. Models of $$\mathsf{CH}+\mathbf{V}{\neq}\mathbf{L}$$. Collapsing $$\aleph_1$$ preserves that $$\aleph_2$$ is a cardinal; the size of the continuum in that model depends on the size of $$2^{\aleph_1}$$ in the ground model. Adding new subsets of $$\aleph_1$$. Brief discussion of the problem of controlling the size of two power sets at the same time. Fourth example class. Example Sheet #4

Last changed: 13 March 2019