/sci/ - Science & Math

Alright guys, here's the deal:
Lately I've been thinking more and more about Godel's incompleteness theorems and other "paradoxical" or unintuitive results in math. Whenever I come across them the general attitude is "yeah, it's unintuitive, but the proof was formed logically and consistently so it's correct". But the more I think of some of these unintuitive results, the more their formulation seems flawed to me.
The main thing most of these results have in common is that they rely on inner paradoxes that are defined by the person who formulated the proof (for instance, Godel shows how to form a statement that is both true and unprovable). But my argument is this:
These paradoxes are always linguistic, not mathematical. They rely on objects and ideas that are not properly defined, in a manner that leaves place for potential contradictions. The only reason these theorems and proof seem correct is that they usually include an example of a self contradictory case (which supposedly supports the proof) whose origin is of a linguistic type. These linguistic paradoxes mostly arise from improper definitions and self reference.
Let's look, for example, at the proof of Cantor's theorem. It's based on defining the idea of a set that contains all elements that are not members of their corresponding subset. But who's to say that this set is even properly defined? Mathematicians are so used to the idea of "let there be..." that they usually just state the existence of objects without first proving that they are well defined and do not lead to possible contradictions.
(part 2 in next comment)

>starting a degree in math and physics this fall
>in the meantime taking a summer course in discrete math for comp-sci majors at a local university
>professor says something about the sum of to rational numbers is always rational. one of the students asks if he can prove it, and the professor says "no, it's an axiom".
>mfw

he makes about 5-6 mistakes every lesson (especially logical fallacies) and everybody just eats it up.
is it like this in every math course for comp-sci majors? what would you do?