TITLE: Understanding Gödel's Incompleteness Theorem
QUESTION [56 upvotes]: I am trying very hard to understand Gödel's Incompleteness Theorem.  I am really interested in what it says about axiomatic languages, but I have some questions:
Gödel's theorem is proved based on Arithmetic and its four operators: is all mathematics derived from these four operators (×, +, -, ÷) ?
Operations such as log(x) and sin(x) are indeed atomic operations, like those of arithmetic, but aren't there infinitely many such operators that have inverses (that is, + and - are "inverse" operations, × and ÷ are inverse operations).
To me it seems as though making a statement about the limitations of provability given 4 arbitrary operators is absurd, but that probably highlights a gap in my understanding, given that he proved this in 1931 and its unlikely that I have found a counter-argument.
As a follow-up remark, why the obsession with arithmetic operators?  They probably seem "fundamental" to us as humans, but to me they all seem to be derived from four possible graphical arrangements of numbers (if we consider four sides to a digit), and fundamentally derived from addition.
[][] o [][] addition and, its inverse, subtraction
[][]
[][] multiplication (iterative addition) and, its inverse, division
There must be operators that are consistent on the natural numbers that we certainly aren't aware of, no?
Please excuse my ignorance, I am hoping I haven't offended any real mathematicians with this posting.

edit: I think I am understanding this a lot more, and I think my main difficulty in understanding this was that:

There are statements that are true that are unprovable.  

Seemed like an impossible statement.  It does, however, make sense to me at the moment in the context of an axiomatic language with a limited number of axioms.  Ultimately, suggesting that there are statements that are true and expressible in the language, but are unprovable in the language (because of the limited set of axioms), is what I believe to be the point of the proof -- is this correct?

REPLY [7 votes]: This is a personal recommendation. 
Peter Smith's "An Introduction to Gödel's Theorems" is very readable and self-contained. 
I believe to fully appreciate Gödel's theorems it is a good idea to learn some computability theory before (Smith does include some of this background in his book, but a bit more of perspective doesn't hurt).
Two very good sources for beginners are Cutland's "Computability" and Boolos and Jeffrey "Computability and Logic". There's a forthcoming book from Enderton (who unfortunately recently passed away), "Computability Theory", which I have been able to read and highly recommend too. For a quick (and unusually clear) introduction to computability theory you can take a look at Enderton's notes, which, by the way, are an excerpt from his book:
http://www.math.ucla.edu/~hbe/computability.pdf
I hope this helps (it did to me).
Edit: Let me also recommend a very nice book by Melvin Fitting, Incompleteness in the Land of Sets. As the title suggests, the author shows us how to discuss incompleteness in its most natural place: the land of hereditarily finite sets.