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Kevin Carlson (aka Arlin) (Mar 22 2024 at 16:43):Julius Hamilton said:
but somehow
Proving that Boolean algebra can be reduced to NAND is a pretty routine exercise with truth tables, not something deep and difficult, if that's the impression you got. You should try it!
Kevin Carlson (aka Arlin) (Mar 22 2024 at 16:44):Hmm, "routine" isn't a good choice of word, there, maybe clearer to say "you could give this as an exercise to CS students taking their first discrete math course."
John Baez (Mar 22 2024 at 17:42):By the way, none of this thread has anything to do with "Glymour’s Theory of Bootstrap Confirmation". If you can axiomatize that theory in ZFC or ETCS or whatever it should be pretty easy for anyone who knows these various approaches to set theory to translate those axioms into any other reasonable version of set theory
John Baez (Mar 22 2024 at 17:42):So I think the real subject header for this thread should be something like "Logic and Set Theory" - and that would get more people interested.
Eric M Downes (Mar 22 2024 at 18:59):(or, if you really want to talk about Gylmour's theory... you might have to do some translation / summary of axioms for us. I bet no one here is very familiar with it? I skimmed some articles and while I think its interesting, didn't see an immediate way to say anything intelligent.)
Julius Hamilton (Mar 23 2024 at 20:15):Eric M Downes said:
I highly recommend Lawvere and Roseburgh Sets for Mathematics for learning ETCS, I've gotten and continue to get a lot out of it.
I’m reading it right now. I think it’s going to blow my mind open.
FYI I changed the title of the thread but am unable to migrate other user’s messages to it. Feel free to migrate one’s own comments if you please. Thank you.
Jean-Baptiste Vienney (Mar 23 2024 at 21:07):I'll suggest you not to invest too much energy in learning ZFC. ZFC is useful if you're doing research in set theory or using set theoretic notions like ordinals, cardinals etc... But there is probably no need of ZFC for talking about "probability, permutations, functions, real numbers" and "a set of individuals".
Looking at Glymour's paper Bootstraps and probabilities, it seems to me that he is talking about inferences in a kind of first-order or maybe second-order (because he talks about subsets) logic, but these inferences are not either provable or unprovable, but have a probability. Maybe some link with fuzzy logic or better probabilistic logic?
Some comment about the relation between ZFC, first-order logic etc...: First-order logic is based on a naive set theory which is not formalized. On the other hand ZFC is a theory in first-order logic. There is this annoying circularity: you need an informal use of sets in order to formalize ZFC. In my mind it is a sign of the impossibility of formalizing everything starting from nothing. You need an informal treatment of something at the beginning.