Comment by edanm
This is fascinating! I haven't read much past the intro yet, but I find the whole premise that you can prove all specific instances of Goodstein sequences terminate at 0 within PA, but not that all sequences terminate (it's a trivial result but still interesting).
I also find it super weird that Peano axioms are enough to encode computation. Again, this might be trivial if you think about it, but that's one self-referential layer more than I've thought about before.
One question for you btilly - oddly enough, I just recently decided to learn more Set Theory, and actually worked on an Intro to Set Theory textbook up to Goodstein sequences just last week. I'm a bit past that.
Do you have a good recommendation for a second, advanced Set Theory textbook? Also, any recommendation for a textbook that digs into Peano arithmetic? (My mini goal since learning the proof for Goodstein sequences is to work up to understanding the proof that Peano isn't strong enough to prove Goodstein's theorem, though I'll happily take other paths instead if they're strongly recommended.)
My apologies for having missed this request.
I don't have good suggestions for a good set theory textbook. Grad school was 30 years ago, and I didn't specialize in logic.
The best set theory book that I read was https://www.amazon.com/Naive-Theory-Undergraduate-Texts-Math.... But that one is aimed at people who want to go into math but do not wish to specialize in set theory, and not at people who want to actually learn set theory.