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Found the meme in the wild. Everything is in the appendix of Emil's thesis.
I think a problem with (but also the amazing thing about) bounded arithmetic is that it is soooo interdisciplinary. Like you need to know a good amount of both proof theory and model theory but also a good amount of theory of computation and algorithms to even begin to understand why people do what they do and why are the problems interesting. And then you're going to borrow methods from basically every algebra-aligned field to figure it out.
And we know nothing. Even though the basic open questions can be formulated somewhat clearly, the things you're going to be working on (especially at a graduate level) are like 5 levels down the rabbit hole and even something like conditional unprovability of one little thing is a state-of-the-art result with huge implications because we don't know how to do anything.
I might have given people the wrong impression... I don't care about cryptography unless you convince me your algorithm would be fun to formalize in PV_1 or tell me how it connects to the strength of the dual weak pigeonhole principle. And sometimes this happens but I still care only when cool logic is happening somewhere in there ;-;
turns out that the reason why i couldn't come up with a proof was because the theorem didn't hold.
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Most of the stuff I've read from my field up to this point has been written by this one guy who's writing is so notorious - very dense, hard to read but genuinely amazing - we have named that style of writing after him. And now when I'm starting to read papers by other people it's genuinely amazing how easy they feel to read but it is a bait - they feel so easy to read they lure you into a false sense of understanding. I guess I just got used to staring at one page for hours before the aha moment comes. This is a reminder that checking the validity of steps in a proof ≠understanding.
Everything in arithmetic is just induction or pigeonhole principle, unfortunately they are not pals.
Induction for only bounded formulas: normal, can't prove the totality of exponentiation but otherwise is pretty reasonably strong
Induction for only sharply bounded formulas: "a power of 2 divisible by 3? after all why not, sounds consistent to me!"
