What are some books on set theory that use formal logic?

[TupoyVolk]

I recently learned predicate calculus from Schaum's Outline of Logic.
in this sort of form:

In addition to refutation trees, however; pfff, refutation trees.

I'm reading "Introduction to Set Theory," Hrbacek, Jech. I'm a little "annoyed" by the informal proofs. Are any books that teach axiomatic set theory through formal logic? Or is that a huge/impossible thing to ask and I'm naively asking for a miracle. The closest thing I can find to what I want is metamath. http://us.metamath.org/mpegif/zfnuleu.html"

I'm familiar with naive set theory, but don't truly see the point (yet) of doing axiomatic set theory without using logic (I know I'll end up doing it anyway because my mind would nag me, though).

Yours faithfully,
A n00b looking for some guidance.

 

[Fredrik]

I'd be surprised if there's a set theory book that proves all the important theorems formally, considering how much work is involved in proving even the simplest things this way. (However, I don't have a lot of experience with different books, so I reserve the right to be corrected by by someone else).

Maybe you should be looking at books that cover proof theory and model theory, not just set theory. For example Kunen's "Foundations of mathematics".

I don't know if it would be such a great idea to prove every theorem formally even if it's doable. I mean, the rules that tell us what a formal proof must look like are just axioms of a proof theory, and they were chosen to ensure that our informal methods of proof can be described formally. So a formal proof isn't a better reason to think that the theorem is true. The only thing a formal proof (of a theorem that can be also proved informally) tells us, is that it can't be ruled out that the axioms or our proof theory are appropriate. (If a formal proof leads to a different conclusion than an informal proof, we would simply conclude that there's something wrong with the axioms of our proof theory).

 

[TupoyVolk]

Thanks for the reply!
I did a few more chapters of Introduction to Set Theory. You're totally right.

I understand it is necessary to prove every single obviously-intuitive things, but they seem pedantic for someone with my goals (I'd like to end up somewhere 'mathematical physics-y'). I think Analysis is where I should be headed.

This may be off topic, but:
I can "do" some differential geometry and Quantum mechanics, however to risk overuse of the chess board metaphor, I only know certain patterns of movements for these things: Not the entire rule of the game. Unfortunately that leads to scary words like 'topology' - which seems foreboding by culture.

So, Analysis: Good direction? I don't want to waste these weeks off.

 

[micromass]

Well, set theory is very interesting, but if your goal is to do physics, then studying set theory is a bit useless.

I think studying analysis and functional analysis is certainly a good thing to do. If you ever will want to study topology and differential geometry, then you will need analysis anyway. So I'd say: go for it

 

[Fredrik]

Thanks for the reply!
I did a few more chapters of Introduction to Set Theory. You're totally right.

I understand it is necessary to prove every single obviously-intuitive things, but they seem pedantic for someone with my goals (I'd like to end up somewhere 'mathematical physics-y'). I think Analysis is where I should be headed.

This may be off topic, but:
I can "do" some differential geometry and Quantum mechanics, however to risk overuse of the chess board metaphor, I only know certain patterns of movements for these things: Not the entire rule of the game. Unfortunately that leads to scary words like 'topology' - which seems foreboding by culture.

So, Analysis: Good direction? I don't want to waste these weeks off.

You didn't give me enough information here. Are you trying to decide which course to take? Set theory or analysis? Analysis is much more useful, but I wouldn't consider time you spend on set theory to be "wasted", because it's always helpful to have a more solid understanding of the foundations.

If you meant mathematical physics (not theoretical physics), you're probably going to have to study real analysis, complex analysis, topology, differential geometry, integration theory and functional analysis, but you can probably do without set theory.

 

[TupoyVolk]

Sorry, it's only self study at the moment. I am in a theoretical physics course at the moment, did 3 years experimental; didn't like it, wanted better understanding. Unfortunately, if I specifically want to go into mathematical physics, it will have to wait until masters.

I'm just deciding what to learn over the holiday break. Three focused weeks can be pretty useful.

Analysis looks like the way to go.

 

[Fredrik]

Yes, analysis is much more useful. Topology is another option. It's annoyingly hard to remember the definitions, theorems and proofs (especially the proofs), so you will have to study the subject many times. It wouldn't be a bad idea to make this the first. (It will make things easier for you when you eventually take a course on real analysis).

I'm trying to learn some functional analysis now, and I think I've spent 80% of the time so far on improving my knowledge of topology. It's quite frustrating. The authors of books on functional analysis are assuming that you know topology really well. (Kreyszig might be an exception, but I didn't know that book existed when I started).