funktour

There are a surprising number of Preview defenders here. You guys must have never had to open a 500+ page document, because for me, that's an all-but guaranteed way to make Preview crash. Preview is only best because the major alternatives (Acrobat) suck more.

PDF readers I actually like: Zathura (obviously), sioyek (if you like customizability and Vim-like bindings, this is a good one!), and Skim.

Everything else tries to do too much (read: be an Acrobat substitute).

(for the record, I think the Penrose take on Gödel and consciousness is mostly silly and or confused)

I think your understanding of the incompleteness theorem is a little, well, incomplete. The proof of the theorem does involve, essentially, figuring out how to write down "this statement is not provable" and using liar-paradox-type-reasoning to show that it is neither provable nor disprovable.

But the incompleteness theorem itself is not the liar paradox. Rather, it shows that any (consistent) system rich enough to express arithmetic cannot prove or disprove all statements. There are things in the gaps. Gödel's proof gives one example ("this statement is not provable") but there are others of very different flavors. The standard one is consistency (e.g. Peano arithemtic alone cannot prove the consistency of Peano arithmetic, you need more, like much stronger induction; ZFC cannot prove the consistency of ZFC, you need more, like a large cardinal).

And this very much does come up for real systems, in the following way. If we could prove or disprove each statement in PA, then we could also solve the halting problem! For the same reason there's no general way to tell whether each statement of PA has a proof, there's no general way to tell whether each program will halt on a given input.

yo let me help you out a little and recommend some resources. here are some series free on youtube: David Spivak and Brendan Fong did a lecture series on their book, Seven Sketches in Compositionality, you can find it on the "Topos Institute" youtube channel. "The Catsters" also have a great introductory series. as far as straight textbooks go, the only intro ones I recommend are 1) Awodey, 2) Riehl, and 3) Seven Sketches.

and one last general tip: categories are simultaneous generalizations of monoids and posets, so whenever you don't understand something, try working it out in the cases where there's only one object (= monoid) or when there's at most one arrow between a pair of objects (= poset)