### G—del's theorem is a mathematical theorem

G—del's theorem is sometimes thought, quite mistakenly, to be less formal or mathematical than other results in logic and mathematics. This is illustrated in the following reflections:
Another difficulty, which is not limited to fuzzy logic, is whether G"odel's proof is actually a proof. To prove incompleteness, we have to interpret the formula and have to understand that what it says is true. That is, the result is not achieved by formal reasoning, but by some meta-reasoning done from outside the system. Hence it is not a ``formal'' proof. It takes insight to see the truth of the formula.
Here the idea that G—del's theorem is not a mathematical theorem in the ordinary sense seems to be prompted by a specific misconception, namely the idea that the undecidable formula constructed in G—del's proof is shown or seen to be true in the course of the proof. If this were the case, one might well wonder how the argument could be carried through in any formal mathematical theory T, since T is shown to be incomplete by the proof, while the sentence shown to be undecidable in T is also shown to be true. However, such is not the case. What is proved is only a hypothetical statement: "if T is consistent, then T is incomplete", and the proof of this statement is a perfectly ordinary mathematical proof, formalizable in elementary number theory. Thus, we need to distinguish between the G—del sentence G, shown (under certain conditions) to be undecidable in T, and G—del's theorem, which is the statement that G is undecidable in T under these conditions. G—del's theorem, but not the G—del sentence, is provable in T.

As proofs go in mathematics (and logic), the proof of G—del's theorem isn't difficult, although some details in the proof of the second theorem (about the unprovability of consistency) are usually dealt with by hand-waving.

### Is there such a thing as G—del's theory?

People sometimes speak of G—del's theory rather than theorem. This is no doubt in many cases just a slip or a sign of unfamiliarity with mathematical terminology, but in other cases there seems to be an idea that G—del did put forward a theory, one that can meaningfully be spoken of as accepted or rejected, established or controversial, and so on. For example:
After all, Peter's theory stands solely on the ABSOLUTE truth of Godel's theory, if it's wrong then his theory would need to be redone
Given that we as human beings have a limited comprehension of the Universe, and that that comprehension will always be less than objective ( as stated in Godel's Theory of Indeterminacy) I beleive that the absolute existence or definition of Evil ( or Good, or anything else , for that matter ) can not be nailed down.
Someone told me that Godel's theory proved most of the axioms that wittgenstein uses in the Tractatus to be wrong. Does anybody have an opinion on this subject?
The implications of this paradox (Richard's) led to things such as Tarski's theory of truth and meta-languages and Godel's theory of incompleteness.
G—del, although he held various controversial philosophical views, did not in fact put forward any theory, and the work in logic for which he is celebrated consisted in introducing concepts and proving theorems, not in anything describable as a theory.