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