Reddit reviews Naive Set Theory

Try Naive Set Theory by Paul Halmos. I think it's aimed at undergraduates, so the content is a bit dense, but the style and tone is very conversational and engaging. I thoroughly recommend it.

Halmos

I don't know where to even start :)

Infinity is a property. Sort of like an adjective. You don't say something is an infinity, but rather something is infinite in size. Think of "infinite" rather than "infinity".

A set is a collection of unordered objects(anything at all), like so

{shoe, car, &, 3}. It's unordered because we can write it as {car, 3, shoe, &}.

A set is said to be finite if we can pair everything in it with another reference set {1, 2, 3, ..., n}.

In the reference set "..." means that numbers continue in the given order. "n" at the end means that there's a number at which the numbers terminate.

Lets call our set {shoe, car, &, 3} A. So, A = {shoe, car, &, 3}.

Now compare the elements(objects) inside A with those inside {1, 2, 3, ..., n}:

shoe can be pared with 1.

car can be pared with 2.

& can be pared with 3.

3 can be pared with 4.

So everything in A is pareable with everything in {1, 2, 3, ..., n}.

So, A is finite according to our definition above.

Definition: S is an infinite set if and only if there exists a set A such that A is a proper subset of S and |A| = |S|.

Ok. This is one of the definitions of infinite set and to understand that you need to be familiar with the notions of functions, mapping, cardinality, bijection, equality, existence, proper subset...all pretty basic notions.

Tell you what, why don't you just study these books below that would teach you all about these notions and much, MUCH more?

A Book of Set Theory by Charles Pinter.

Naive Set Theory by Paul Halmos.

The books above will not only teach you about finite/infinite sets, but also can serve as a very nice foundation to study higher math.

Translate Naive Set Theory to Swedish, it's only around a hundred pages so you could make it in time.

Set Theory:

Naive Set Theory

Number Theory:

Elementary Number Theory

Introduction to Analytic Number Theory

A Classical Introduction to Modern Number Theory

Topology:

Topology

Introduction to Topological Manifolds

If you're completely new to the subject, I would pick up an introduction to proof, such as Velleman's How to Prove It. They all have a chapter featuring a good introduction to set theory.

If you're up for something a little bit, but not too much, more challenging, you could pick up Halmos' Naive Set Theory and learn a bit about the axioms underpinning set theory.

Sadly, I can't think of a title in discrete math or introduction-to-proofs that I can recommend. A common recommendation for the latter category, which I haven't read myself but has a good reputation, is the following:

• How to Prove It: A Structured Approach, 2nd edition by Daniel J. Velleman.
  
  Another book which has a good reputation is
  
  
• How to Solve It: A New Aspect of Mathematical Method by George Pólya.
  
  The book even has its own Wikipedia article!
  
  ---
  
  These, however, are both about proofs as their own technique. I wish I could provide a recommendation for books on discrete math, introduction to set theory, and the related topics I mentioned above. You might consider something like
  
  
• Naive Set Theory by Paul Halmos
  
  (This title also has its own Wikipedia article, too.)
  
  but I'd defer to others for recommendations on textbooks for these prerequisite concepts and principles useful to an analysis student.
  
  ---
  
  >Also, when should I start my real analysis? Can i study it with the calculus or after completing calculus?
  
  I'd consider taking real analysis after completing the introductory sequence in calculus, possibly including multivariable calculus, linear algebra, and an introduction to differential equations. I'd also wait until after you've had a good introduction to mathematical proofs, something most universities and colleges present in a class on discrete mathematics.
  
  If you jump into analysis completing at least one-variable differential and integral calculus, as well as a class with a strong proof-based component, you're likely to find yourself in over your head.
  
  First, most analysis classes assume the students are already familiar with ideas like convergent sequences, limits, continuity, differentiability, and integration. This is all presented again in a much more rigorous way, but it's typical for an analysis class to lean heavily on prior interaction with such topics.
  
  Second, analysis is a heavily proof-based class, and learning how to read, understand, and write proofs is its own skill set. Trying to acquire fluency in proofs by taking an analysis class, despite no prior formal encounters with proofs, will make analysis considerably more challenging for you.
  
  I hope this helps some. Good luck!

I've always heard Naive Set Theory by Halmos is good. Note, that it isn't actually about naive set theory, but axiomatic set theory

http://www.amazon.com/Naive-Set-Theory-Paul-Halmos/dp/1614271313