# Looking for book on ZFC axiomatic set theory (I think)

I am looking for a book that starts at the standard ZFC axioms and progresses to the point where some recognisable non-trivial mathematical statement is proved. By recognisable I mean something that you may encounter in school/early university level and is not purely set-theoretical (e.g. infinitely many primes, triangle inequality, etc.).

Can anyone recommend a text or combination of texts to me if such a collection exists? Thanks.

fresh_42
Mentor
Why do you want to start with ZFC and then not end in set theory?

The examples you gave can be proven and usually are without explicitly mentioning ZFC. And the C of ZFC isn't needed there.

I just want to understand the process from the ground up - kind of like in programming knowing assembly can give you some insight into coding with higher level languages, though this is more for interest than anything else. I expect I will learn a bit (possibly a lot) of set theory in the process. I guess the end goal is so that I can prove other theorems using the same techniques. I have studied a lot of maths, have a degree in it, etc. but never studied set theory formally in depth and it would be good to fill in some of the gaps in my knowledge. So if you have any texts you could recommend that would be much appreciated.

fresh_42
Mentor
I'm afraid not. I have a nice book on set theory which is more entertaining rather than purely theoretical although it was written by a mathematician. It shows the different kinds of infinity or how a square can continously filled by a line and stuff like that. I've read about the work of Bertrand Russell on occasion. And I know a proof of the equivalence of several statements to the axiom of choice, e.g. Zorn's Lemma. But this is more an appetizer in book about abstract analysis.
I also know that there are branches in mathematics which deal with multiple valued logics or constructive logic (I think they call it, i.e. not allowing indirect proofs.) But I've not been into. I love my naive and intuitive ZFC approach.

The reason I was asking is, I guess there are books who start with this axiomatic approach but end up in either pure set theory, logic or philosophy. And there are certainly books like the one I mentioned drifting into a more entertaining direction.
This one for example: https://www.amazon.com/Gödel-Escher-Bach-Eternal-Golden/dp/0465026567/ref=sr_1_1?s=books&ie=UTF8&qid=1447367427&sr=1-1&keywords=goedel+escher+and+bach

At least I don't know any of the kind you're looking for, sorry.
And I doubt a little that it will provide you the insights you want to gain. It's a bit like learning proto-Indo-European in order to learn more easily English, French and Swedish.
But honestly: Why the heck do you want to know stuff as assembly? That reminds me on a prof who when the university changed its main computer let them put the old one into his office. He then held courses where you could learn how to do programming be literally switching single bits in the upper address room. Nice little red lights showed you which bit was set. Looked like on the bridge of the enterprise in Star Trek- the original series.

I have the first of a three-volume set of U of I, that uses naive set theory to construct arithmetic, etc. You have any interest in a pointer to that?

@aikismos That couldn't hurt, thanks.

@fresh_42 In programming, even if you never write assembly, it can have a practical use to know how things compile for optimisation purposes. Not really applicable for maths though since the end result is in most cases the main point, rather than how elegant/optimised the proof is. As I said before, it's mainly out of curiosity that I wish to know.

fresh_42
Mentor
@fresh_42 In programming, even if you never write assembly, it can have a practical use to know how things compile for optimisation purposes.
That's right. Plus there are a hell lot of mainframes out there which still run on programs of the 70's and 80's. And many of them can't hardly be substituted.

Not really applicable for maths though since the end result is in most cases the main point, ...
A fat appeal here! Almost all mathematicians can distinguish between a good and a bad proof, i.e. a beautiful one or one from a lumberjack. And there is a common belief that non elegant proofs have a good chance to fail somewhere. Mostly it's true. Only good proofs reveal the inner beauty of math. Think of Euler's formula ##e^{iπ} = -1## or the many formulas and occurrences of ##π## in nature.

I agree with everything you've said, but in general it is the result that advances mathematics, whereas very few proofs open up genuinely new fields of study or do things in a truly revolutionary way. For every proof like (for example) Wiles' of FLT (which is way over my head, but apparently introduced several new concepts), there are probably 100,000 that just get the job done without changing the way people do or think about maths.

fresh_42
Mentor
I agree with everything you've said, but in general it is the result that advances mathematics, whereas very few proofs open up genuinely new fields of study or do things in a truly revolutionary way. For every proof like (for example) Wiles' of FLT (which is way over my head, but apparently introduced several new concepts), there are probably 100,000 that just get the job done without changing the way people do or think about maths.
Yep, probably true. And don't bother Andrew Wiles. You won't find many on this globe who understand the complete proof. And if you try, I beg you not to start at ZFC.

JJ1979
Yep, probably true. And don't bother Andrew Wiles. You won't find many on this globe who understand the complete proof. And if you try, I beg you not to start at ZFC.
Especially since Wiles doesn't work in ZFC!

fresh_42
Mentor
Especially since Wiles doesn't work in ZFC!
Anyone claimed? But I bet the C is part of the proof.

@JJ1979 The title: Set Theory and the Structure of Arithmetic by Hamilton and Landin (Allyn and Bacon)
no ISBN, but Library of Congress Catalog: 61-15038

Using a sufficiently axiomatized set theory (naive), the three volume set establishes all sorts of proof by building up the natural numbers (through Von Neumann's construction) through the reals (by Cantor sequences) and then in the two latter volumes explores abstract algebra and Euclidian geometry through a set-theoretic framework.

Thanks aikismos, this looks very interesting. I found a pdf of the book online here.