Thanks for pointing this out. Two questions: (1) Are there any resources you'd recommend reading to learn more about this? (2) If you are familiar, could you restate the Incompleteness theorem in your own words? Just looking to learn more here :)
1) Smith's Introduction to Godel's Theorems http://www.logicmatters.net/igt/ is a great book, with all the mathematics but willing to go into the philosophy.
2) Franzen's Gödel's Theorem: An Incomplete Guide to Its Use and Abuse http://www.ams.org/notices/200703/rev-raatikainen.pdf is enlightening in a different way.
I'll also mention a favorite author around here http://www.scottaaronson.com/blog/?p=710 .
One goal was to come up with a collection of axioms for the Natural numbers (capital letter to denote the Natural numbers we all know and love). What was wanted was a collection of axioms that are recursively enumerable. Think computable. The goal was to find a mechanistic process to check theorems and to prove new theorems. In some sense too alleviate the field from human error. In modern language we'd say to find a way to have a computer check/discover theorems in number theory.
There are two axiom systems for the Natural numbers. Both are Peano axioms and one system is first order and recursively enumerable. The other system is second order and not recursively enumerable. There are infinitely many models of the first order Peano axioms. For each such model we call them natural numbers (lower case) to signify they are a model of the first order axioms.
The Incompleteness Theorem: There are statements that are true in the Natural numbers (upper case!) that are not true in all models of the natural numbers. Furthermore, this will always be the case no matter what system of recursively enumerable axioms you have for describing the Natural numbers.
Consequence: The Natural numbers can not fully be described by a nice set of axioms. Whatever recursively enumerable system of axioms you have to describe the Natural numbers will be insufficient to prove all true statements of the Natural numbers. Hence, such systems of axioms are incomplete.
One can always find a complete system of axioms by taking as the collection of axioms the collection of all true statements. This isn't helpful because there would no effective way to determining whether or not a statement is an axiom just by looking at it or comparing it to a finite set of axiom schema. Such a collection of axioms is wholly impractical and not useful. But it is wrong to say that a complete axiom system can not be found.
Some people falsely claim that the Incompleteness theorem says that there are statements of the Natural numbers that are neither provable or disprovable. What the theorem says is that for a given recursively enumerable set of axioms that the Natural numbers are a model of there will be statements of the Natural numbers that are true but not provable in that system. All true statements of the Natural numbers are provable in the second order system of axioms but the things get dicey from a logic point of view when working with the second order Peano axioms.