2019: Permutation Models

Here are the recorded lectures from the Mastermath course in Set Theory (autumn 2019) about permutation models and the consistency of the negation of the Axiom of Choice. (Also available for binge-watching in a virtual box-set on YouTube.)

2019-11-18: Introduction

Set theory with atoms; permutations and symmetric sets. The first Fraenkel model.

• First hour
• Second hour
• My notes

2019-11-25: Three models

The Basic Fraenkel model: an infinite Dedekind-finite set. The Second Fraenkel model: a sequence of pairs without a choice function. The ordered Mostowski model: a model where AC fails yet every set admits a linear order.

• First hour
• Second hour
• My notes

Here is Russell's paper, which contains his "Paradox" (p. 32) and the example of the pairs of boots (p. 47).

2019-12-02: Global linear order; how to make the models

Proof that the ordered Mostowski model has a global linear order. How to make models with atoms using ZF(C).

• First hour
• Second hour
• My notes

2019-12-09: How to make models for ZF without AC.

The Jech-Sochor embedding theorem with a very superficial description of forcing. Start of the proof that in the ordered Mostowski model there is no `cardinality function'.

• First hour
• Second hour
• My notes

2019-12-16: Last lecture

Finishing the proof started in the previous lecture and indeed the course

• First hour
• A few extra seconds
• Second hour
• My notes