I have heard a great deal about the "incompleteness theorems" of Godel, but only in vague terms. I have often wondered how we can be certain that whatever axioms we may accept for a theory will not lead us into a contradiction somewhere, perhaps one that is very subtle and hard to detect. Similarly, I have wondered if it may be possible to both prove and disprove something with the same theory. It may seem completely self evident that this cannot happen, but I would be interested if it could be proved that it cannot happen. However, the entire point of Godels thought may be that such a "metaproof" would then be relying on some higher theory to reason about our original theory, and that this higher theory, too, is subject to the same limitations as our original theory.Does Godels theorem have anything to say about this?
Similarly, as mathematics becomes more complicated, abstract, and towering, we must admit that perhaps even the sharpest minds will fail at some point (like andrew wile's first attempt to prove fermats last theorem) and we might all become convinced of the truth of something that is "really" wrong. Perhaps in some cases (and perhaps this has already taken effect) it will take a very long time before we find a problem and even longer to find the root of the problem. In this way, cannot even the most rigorous theory be regarded as a hypothesis, even within the context of the theory?
My last question pertains to geometry. It can be phrased in a few different ways, but it centers on one idea. When I learned that integrals could be used to calculate curved volumes, I wondered (and still do) if there exist curves that can be rotated and added together piecewise so as to find the volume of any conceivable arbitrary volume.

Mathematics is a mental game, and it is not required to exactly represent absolute validity.
Arithmetic is the most concrete, calculus the least, IMO.
Application of mathematical concepts to the analysis of reality is where the problem of "incompleteness" arises.

For instance take the parameter 'time', this concept like math is useful for analysis, but also like math it does not really exist.
There in lies the enigma. Its all in the mind, you know !

IMU

I do not even mean to suggest bringing the discourse to the subject of applying mathematics to reality; I suppose my central concern is with internal consistency.

Alyosha: The Goedel Proof esentially states that either there is a valid but unprovable theorem or else mathematical logic is inconsistent. Nobody is willing to accept the view that mathematical logic is inconsistent, and there is good reason to reject this view.

Hence the Goedel Theorem is viewed as proving incompletelness or some basic limitation in what can be proven.

A theorem which is valid but unprovable can be added to the axioms of the system, expanding the scope of the system and introducing some additional valid but unprovable theorems (such theorems would not be considered appropriate for for the original system).

I think that the above is one variant of a layman's view of the Goedel Theorem, suitable for explaining to others like myself who do not have the background to cope with the actual Goedel Proof and its implicaitons.

Somebody who understands the theorem better than I might take exception to my explanation and have good cause to do so.

The Goedel Proof is viewed as closing the book on a quest for a proof of the consistency of the axioms of mathematics.

This quest started perhaps 150 years ago when Riemann attempted to prove the parallel postulate of plane geometry. His plan was to assume that the postulate (or axiom) was invalid and attempt to prove some paradoxical theorem, resulting in a Reductio Ad Absurdum proof of the parallel postulate. One result of his efforts was a consistent set of theorems valid on the surface of a sphere, and the development of differential geometry.

Another result of his efforts was the initiation of the quest for a proof of the consistency of the axioms of geometry and other mathematical systems. Along the way to the Goedel proof, mathematical logic became more formal, discarding various “sloppy” intuitive proofs, which resulted in the acceptance of proofs of some invalid theorems. Prior to logic becoming more formal, some of Cantor’s work might not have been accepted due to being counterintuitive, even though formally correct. Id est: "Sloppy" proofs might have been accepted as refuting Cantor's work in the ear when logic was less formalized. .

I have wondered if it may be possible to both prove and disprove something with the same theory.

You should do some research on the concept of dialetheism. You are not the first to ask such questions.

Similarly, as mathematics becomes more complicated, abstract, and towering, we must admit that perhaps even the sharpest minds will fail at some point (like andrew wile's first attempt to prove fermats last theorem) and we might all become convinced of the truth of something that is "really" wrong.

You are asking here what we mean by "proof". To the ancient Greeks proof is totally alien to what we now think of proof. Indeed, a mathematician in the time of Newton would have little idea of the concepts underlying the proofs of modern mathematics. Less than 400 years ago geometry was everything. A proof was either geometrical or it was nothing. The idea of proof is an ever evolving concept. Great mathematicians such as Euler and Lagrange gave proof s of theorems that by todays standards fall way short of proof. In fact Euler used many faulty ideas in some of his proofs, making them downright wrong.
I guess the moral of the story is, at the end of the day proving a theorem is just convincing yourself and your peers that the socalled theorem is true. What else can a proof possibly be? I would imagine that modern mathematics, at the rate in which new papers are written, is full of faulty proofs, incomplete proofs and wrong proofs, and possibly even theorems that are false. The concept of proof, the rigor of proof and the means of proof constantly change, as our idea of mathematics changes.

I suppose what I think of, when I think of "proof", is a theory in which we accept some things as self evident, along with definitions of objects and operations that make sense within the context of the theory. Theorems must be proven in a manner that shows something implies something else in the (seemingly) inescapable manner of "All men are mortal, Socrates is a man, Socrates is mortal", and that these implications are entirely the result of our earlier axioms, definitions, and theorems. Note that none of this implies that the theorems are true in any absolute sense, but only in the sense that "if all men are mortal" and "if socrates is a man". I do not even mean to imply that the proof of Socrates mortality is "absolutely inescapable", but in the context of our theory we may accept it as inescapable.