und aus dem Chaos kam eine Stimme, und die Stimme sprach zu mir
"Lächle und sei fröhlich, denn es könnte schlimmer kommen."
und ich lächelte und war fröhlich
und es kam schlimmer.

I gave a short talk about the irrefutability of the axiom of choice from ZF for some younger students, in the scope of a seminar of the best mathematics club of the multiverse.

I wanted to create a script, and instead of putting up a LaTeX-document which will then collect dust in one of my backup directories, I thought I'd write a short series of postings giving a script for this talk. Unfortunately, it is rather complicated, and I am not sure whether I can recommend that.

The talk was in German, these posts will be in English, though. I will separate them in parts, and will publish the parts delayed, as they will be very long. I hope that everyone interested in it will enjoy it.

In the end, the talk was a lot shorter and I have left out a lot of stuff. It is impossible to put all of that matter into a two hours talk for people not familiar with logic.

It is based on my lecture notes for the lecture "Models of Set Theory" given by Donder (whom I thank for giving his permission to do so), and the book "Set Theory" by Kunen, and "Einführung in die Mengenlehre" by Oliver Deiser.

What I want to do is give an ontological introduction to the topic, as I think there is no such thing yet. That is why I do not give the proof for the independence of AC, but only for the irrefutability.

This is not a scientific paper!

This time I will only publish an outline, which I will fill with hyperlinks as soon as I post them (so this post will be updated until all parts are published):

Part 1 will be about the basic logical background needed for this:
  • Formulae
  • First-Order Theories
  • Interpretations and Models
  • A Simple Independence Proof
Part 2 will be about ZF:
  • Classes
  • Axioms of ZF
  • The Axiom of Choice
  • Well-Orderings, Ordinals and Transfinite Induction
Part 3 will be about inner models:
  • The V-Hierarchy
  • Inner Models
Part 4 will then contain the actual proof:
  • The Classes OD and HOD
  • HOD \models AC
It is likely that I made some mistakes, so if you see one, feel free to send comments.