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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.