Topics in Set Theory
Part III of the Mathematical Tripos
Lent Term 2018

Lecturer: Professor Benedikt Löwe, email:
Schedule: Lectures: Monday, Wednesday, Friday, 11–12, MR13.

Example Classes:
#1: Thursday 1 February, 3–5:30, MR13.
#2: Thursday 15 February, 3–5, MR13.
#3: Thursday 1 March, 3–5, MR21.
#4: Wednesday 14 March, 3–5, MR21.

Revision Session: Wednesday 23 May, 3–5, MR21.

Friday, 19 January 2018 First Lecture. The Continuum Problem and its history. General remarks about the incompleteness phenomenon and the desire to have nice models of set theory to do model-theoretic constructions. The von Neumann hierarchy, Mirimanoff rank, basic properties, and closure properties of limit levels of the von Neumann hierarchy.
Monday, 22 January 2018 Second Lecture. Failure of the axiom of Replacement in the von Neumann hierarchy at singular and successor cardinals. Proof that successor cardinals are regular. Regular limit cardinals and inaccessible cardinals.
Wednesday, 24 January 2018 Third Lecture. The Aleph and the Beth hierarchies, aleph and beth fixed points. ZFC at inaccessible limits of the von Neumann hierarchy. First proof that ZFC does not prove IC (using Gödel's Incompleteness Theorem). Tarski-Vaught test. Building the Skolem hull inside Vκ to get a (not necessarily transitive) countable model of ZFC.
Friday, 26 January 2018 Fourth Lecture. Example of absoluteness breaking down for non-transitive models: the formula describing the number one may not be absolute. Mostowski collapse and the existence of countable transitive models of set theory. Construction of worldly cardinals that are not inaccessible. Absoluteness of atomic and quantifier-free formulas. Observation that hardly any formulas are quantifier free.
Monday, 29 January 2018 Fifth Lecture. Bounded quantification and the class of Δ0 formulas. Absoluteness of Δ0 formulas for transitive models of set theory. List of set theoretic concepts that are Δ0.
Wednesday, 31 January 2018 Sixth Lecture. Absoluteness continued: examples of formulas not absolute for transitive models ("x is a cardinal", "x is regular"); upwards and downwards absoluteness; Σ1 and Π1 formulas. Second proof of the non-provability of IC from ZFC. Note of caution about the use of absoluteness in the earlier arguments about axioms of set theory in the levels of the von Neumann hierarchy.
Thursday, 1 February 2018 First Example Class. Example Sheet #1.
Friday, 2 February 2018 Seventh Lecture. Absoluteness of wellfoundedness. Absoluteness of formulas defined by transfinite recursions via absolute formulas. Relativisation of formulas. Definability and its paradoxes. Internalised notion of definability.
Monday, 5 February 2018 Eighth Lecture. Ordinal definability and the class OD. Closure of OD under Pairing. Well-ordering of OD and the Axiom of Choice in OD.
Wednesday, 7 February 2018 Ninth Lecture. The question of the transitivity of OD. The hereditarily ordinal definable sets. ZFC in HOD.
Friday, 9 February 2018 Tenth Lecture. Cancelled due to illness.
Monday, 12 February 2018 Tenth Lecture. Summary of our various attempts to build inner models of CH: truncation, countable transitive models, HOD. Non-absoluteness of the set identified by ordinal-parameter definition. The definable subset operation. The constructible hierarchy and its basic properties.
Wednesday, 14 February 2018 Eleventh Lecture. The ordinals in L. ZF in L. Non-absoluteness of the definition of HOD vs absoluteness of the definition of L.
Thursday, 15 February 2018 Second Example Class. Example Sheet #2
Friday, 16 February 2018 Twelfth Lecture. Minimality of L. Consistency of V=L, V=HOD and V=OD. Axiom of Choice in L. The Condensation Lemma (with proof sketch) and CH in L.
Monday, 19 February 2018 Thirteenth Lecture. Proof of the condensation lemma. The method of inner models cannot be used to provide models of the negation of CH. Relativised constructible universes and the idea of adding new sets to a countable transitive model of set theory.
Wednesday, 21 February 2018 Fourteenth Lecture. Forcing partial orders, compatibility, density, filters, generic sets. The partial order of finite functions from x to y. Generic sets for that partial order define a surjection from x to y.
Friday, 23 February 2018 Fifteenth Lecture. Existence of generics for countable collections of dense sets. Generic sets for a model of set theory. Non-existence of generic sets for splitting partial orders. Recursive definition of names (Boolean case) and simple examples of interpretations of names.
Monday, 26 February 2018 Sixteenth Lecture. Partial order version of names; canonical names; valuation of names; name for the filter. Definition of M[G] and Minimality Theorem for M[G]. Proof of the Pairing Axiom in M[G].
Wednesday, 28 February 2018 Seventeenth Lecture. A first attempt to prove the Power Set Axiom. The forcing language. The (semantic) forcing relation. Statement of the Forcing Theorem (no proof).
Thursday, 1 March 2018 Third Example Class. Example Sheet #3
Friday, 2 March 2018 Eighteenth Lecture. The syntactic forcing relation. Restatement of the Forcing Theorem in terms of the syntactive forcing relation. Proof of the power set axioms under the assumption of the Forcing Theorem.
Monday, 5 March 2018 Nineteenth Lecture. Proof of the Forcing Theorem: cases conjunction, negation, existential quantification, and atomic formulas with element symbol.
Wednesday, 7 March 2018 Twentieth Lecture. Proof of the Forcing Theorem: case of atomic formulas with equality symbol. Relationship between syntactic and semantic forcing.
Friday, 9 March 2018 Twenty-first Lecture. Generic model theorem: M[G] satisfies the axioms of ZFC: Separation, Replacement, and Choice.
Monday, 12 March 2018 Twenty-second Lecture. Height of M[G]. Minimality Theorem. Consistency of VL. Preservation of cardinals. Countable chain condition. The countable chain condition preserves preservation of cardinals.
Wednesday, 14 March 2018 Twenty-third Lecture. Countable chain condition of Fn(X,ℵ0). Δ systems. Δ system lemma (no proof). Proof of the consistency of ZFC+not CH. Fixing the size of the continuum: nice names.
Wednesday, 14 March 2018 Fourth Example Class. Example Sheet #4

Last changed: 29 May 2018