Reddit reviews Foundations of Analysis (Ams Chelsea Publishing)
We found 6 Reddit comments about Foundations of Analysis (Ams Chelsea Publishing). Here are the top ones, ranked by their Reddit score.
My favorite book: http://www.amazon.com/Foundations-Analysis-Graduate-Studies-Mathematic/dp/082182693X
Available online at: http://www.futuretg.com/FTHumanEvolutionCourse/FTFreeLearningKits/01-MA-Mathematics,%20Economics%20and%20Preparation%20for%20University/006-MA06-UN01-05-Analysis/Additional%20Resources/Edmund%20Landau%20-%20Foundations%20of%20analysis.%20The%20arithmetic%20of%20whole,%20rational,%20irrational%20and%20complex%20numbers.pdf [PDF]
You can construct the naturals, integers, rationals, reals, and the complex numbers all in terms of sets. The constructions for everything except the reals are elementary, and the reals aren't too hard, just more involved. There's a short book by Landau that does all of these, you should check it out.
Cardinals are defined in terms of ordinals, which are defined in terms of order types and well ordered sets.
Most things that you will deal with on a regular basis can be described in terms of sets. However, due to Russel's paradox, sometimes we want to talk about things that can't (consistently) be considered sets. These objects often show up in category theory, often as objects that are "too big" to a set (see proper class).
I'm sure someone who knows more about category theory than I do can give you lots of example of categories that aren't sets.
You need some grounding in foundational topics like Propositional Logic, Proofs, Sets and Functions for higher math. If you've seen some of that in your Discrete Math class, you can jump straight into Abstract Algebra, Rigorous Linear Algebra (if you know some LA) and even Real Analysis. If thats not the case, the most expository and clearly written book on the above topics I have ever seen is Learning to Reason: An Introduction to Logic, Sets, and Relations by Nancy Rodgers.
Some user friendly books on Real Analysis:
Some user friendly books on Linear/Abstract Algebra:
Topology(even high school students can manage the first two titles):
Some transitional books:
Plus many more- just scour your local library and the internet.
Good Luck, Dude/Dudette.
Yes, for math there is Edmund Landau's Foundations of Analysis. Read the reviews.
Start With "Foundations Of Analysis" By Edmund Landau
http://www.amazon.com/Foundations-Analysis-AMS-Chelsea-Publishing/dp/082182693X
It's a tiny book, but is very good at explaining basic abstract algebra.
Here is the description from Amazon:
"Why does $2 \times 2 = 4$? What are fractions? Imaginary numbers? Why do the laws of algebra hold? And how do we prove these laws? What are the properties of the numbers on which the Differential and Integral Calculus is based? In other words, What are numbers? And why do they have the properties we attribute to them? Thanks to the genius of Dedekind, Cantor, Peano, Frege and Russell, such questions can now be given a satisfactory answer. This English translation of Landau's famous Grundlagen der Analysis-also available from the AMS-answers these important questions."
With the above book you should then have enough knowledge to move on to calculus.
I recommend the two volume series called "Calculus" by Tom M. Apostol.
The first volume is single variable calculus and the second is multivariate calculus
http://www.amazon.com/Calculus-Vol-One-Variable-Introduction-Algebra/dp/0471000051/ref=sr_1_4?ie=UTF8&s=books&qid=1239384587&sr=1-4
http://www.amazon.com/Calculus-Vol-Multi-Variable-Algebra-Applications/dp/0471000078/ref=sr_1_3?ie=UTF8&s=books&qid=1239384587&sr=1-3
I'm currently on this journey as well! I'm a programmer teaching my self rigorous maths, so I can definitely help you out.
I find it's best to simultaneously look at several resources on topics such as proofs, so you can get a few perspectives on the same essential topics and have an easier time of finding something.
As a preliminary to proofing, I would suggest a survey of basic logic and Set Theory. I picked up my Set Theory from google searches and the introduction in Apostol's Calculus, and wiki articles on logic and set operations.. It's really easy to learn enough set theory and logic to begin understanding rigorous proofs.
To learn the proofing skills needing for Real Analysis I recommend
a) "Foundations of Analysis" by Edmund Landau.
b) Math 378: Number Systems: An Axiomatic Approach
For an actual book on real analysis, there can be no greater book than Apostol's Calculus.