I'm getting into formal and rigorous mathematics and I'm looking for a book on proofs, axiomatic systems, and set theory. My only exposure to axiomatic systems proper was in a rigoros euclidean geometry book, and set theory was breifly explained in baby rudin, but I'm looking for a book that deals more thorougly with these subjects.
Pic is what I've been looking at, but I'm not sure if it's any good.
Also as a sidenote, am I correct in my understanding that "axiomatic system" is basically what we mean when we say "rigour" in mathematics? An axiomatic system and the logical steps we use to derive theorems from those axioms? Is there a major difference to the systems (except for of course the individual axioms, I'm talking methodology here) we use today and the one Euclid developed?
A (the only comprehensive afaik) contemporary overview on introductory literature to your matters is
A 100 page overview, including reviews/comments on the classic books.
>am I correct in my understanding that "axiomatic system" is basically what we mean when we say "rigor" in mathematics?
>Is there a major difference to the systems [...] we use today and the one Euclid developed?
Euclid didn't know Frege and Descartes (use quantifiers, introduce coordinates, give her the D, and other acts of violence).
Today shit is more formal than before, and you have different systems also (compare Hilbert style and Gentzen style proof theory). But derivation rules like "from A and A=>B being true, we conclude B is true" were and are at the core, if that's what you mean.
That's a really nice text, anon.
So should I try to look into books on set theory and logic separately? I noticed the text does give a number of example works, if I go through the text and read through the books recommended (As well as maybe reading an intro to proofs book), would I be well prepared for the more rigorous parts of mathematics?
If you describe your motivation and where you want to go and why, I might advise you better. Both logic and set theory are infinite holes and most few both as tools to do what they actually want to do and know. Where do you stand there? Do you want to know logic per se? And why? Most texts of elementary set theory do introduce logic anyway. Most text on topology introduce set theory in the same so-so fashion. Most texts of differential geometry will introduce topology like that, and books on physical fields will introduce differential geometry like that (or in a raw coordinate dependent form). See my point?
I personally like the notes here
which are on classical logic and end with formalizing set theory. But if some anon recommends a classic book on set theory, this will most likely do a good job for you as well.
The hard and also interesting part in doing all that stuff is forming your own opinion what math is and what it can do.
>If you describe your motivation and where you want to go and why
Well my motivation is that I've been reading more rigorous mathematical textbooks. I read both baby ruding and a text called "Basic Geometry" but even though they both were formal and rigorous texts they seemed to handle the idea of "rigour" differently. This was confusing to me, so I decided I'd try to learn about set theory and logic, formalism, and axiomatic systems separately so I'd be better prepared for formalism in mathematics. I'm a hobbyist mathematician at heart, not a philosopher.
Is reading a proper book on set theory worth it? I've basically picked set theory up on the go to the extent that I can understand axiomatic probability theory (e.g. Casella & Berger, a god-tier book btw) and such. I'm also familiar with the difference between finite and infinite sets (Cantor).
Nothing complex though.
Then you can try to read the first 77 pages of the first link in
i.e. this pdf
and get to know what formal proofs are or in which sense they are math and they work,
and I assume you'll be smarter in the end but also see why books like Rudin don't operate this way, i.e. why math by mathematicans is done as it is.
(That's what I expect, but of course I hope you stick with it and eventually do away with mess that is contemporary mathematics)
I like Suppes. It's not the text my university uses but it's far more rigorous and formal (I haven't seen any other sources as explicit and precise).
That said, I would NOT recommend diving into this if you haven't done any formal logic first. Not that it's impossible to do without logic but you will get less out of it and you will struggle more.
from easiest to hardest:
How to Read and Do Proofs: An Introduction to Mathematical Thought Processes by Solow
How to Prove It: A Structured Approach by Velleman
Book of Proof by Hammack (Free: http://www.people.vcu.edu/~rhammack/BookOfProof/)
A Transition to Advanced Mathematics by Smith, Eggen, and Andre
A Primer of Abstract Mathematics by Ash
Conjecture and Proof by Laczkovich (a follow up to the above)
I strongly disliked the method of exposition in Hammack. Everything is in short, clipped sentences that get repeated a few times in a row. Maybe he's trying not to be overwhelming or terse but it feels like he's trying to teach his cat rather than a functionally intelligent person.
The exercises were mostly pretty bland and elementary too.
Velleman is a much better introduction imo.
I don't have an experience with Suppes, but I took an axiomatic set theory course and we used Enderton and then Jech. I liked Enderton, he has also written books on mathematical logic and Recursion Theory.
If your looking into being very formal, you should probably study some metalogic. I mean go through a text that developes completeness and decidability.
get the book on the left: you will learn baby set theory done in category theory in passing.
Perseverance, I guess. It's worth to note however, that it's really hard to find anything about proofs, or set theory, or mathematical logic anywhere for self study guides. Most people just tell you to learn calculus then go to baby rudin.