Used to analysis/topology

> you use mathematical tools to essentially prove things about mathematics as a whole
>prove things about mathematics as a whole
But it's not what you do. It's like saying that mathematical physics proves things about the real world. Mathematical logic defines structures that look like mathematical objects we use everyday, then studies these things instead.

So you want a theory of logic developed from first principles?
Well, your meta-theoretic first principles can't make use of logic or any logical arguments at all, otherwise you'd be guilty of circular reasoning by assuming what you set out to prove (which is, formally, a logical fallacy).

In other words, your first principles of logic must necessarily be illogical, and unmathematical as well (since logic is a precursor to mathematics). And as Tarski's undefinability theorem (the analogue of Godel's theorem for truth) shows, you can't get out of it by assuming "metalogics all the way down".

I'm not saying that what you're asking for is impossible (after all, you can never completely enumerate all the natural numbers, but that hasn't stopped mathematicians from developing a theory of infinite sets) but it's going to require something that would be unrecognizable as mathematics or logic to most of us.

>used to logic/analysis
>start studying topology and differential geometry
>get fiddled in the butthole

How are you used to logic and analysis but not to topology?

>Number theory is pretty easy to understand tbqh
>nobody can even find zeros of the zeta function for 150 years
>still no way to figure out how many rational points are on an elliptic curve

This is an unusual advice, but:
If you want into logic try doing as many proofs by yourself as possible! Even the ones given in the textbook. (Btw.: It gets worse. Hilbert-System for first (and higher) order logic can be a real pain in the ass.)

>Used to graph theory (so think algorithms are easy; also Veeky Forums told me that everything in CS is easy).
>Start studying algorithms on character strings (yes this is actually a whole research field).
>bamboozling experience.
>mfw

Yours is a sound advice since I learnt a couple new things I had missed, despite having spent a lot of time figuring out theory.

I solved this in three sittings today.
It took me ~7h15min in total.
Is this normal?

I may have transcribed something incorrectly but I think the idea is overall correct.
I wonder if that's how it's supposed to be done.

Well, for a complete beginner it is normal.
But after a half year or so - after you've learned systems like resolution, hilbert, sequence (Genzen), etc. - it should take much less time.
In my logic exam we had a similar thing to prove in 10 minutes.
After this time, it should be as easy for you, as it is for computer to compute an algorithm.