### Gödel's theorem without truth

The concepts of consistency and completeness are purely syntactical concepts, meaning that they are all about formal rules and formulas, and don't involve any notion of truth or falsity. More precisely, that a theory T is consistent means that there is no formula A such that both A and its negation ~A have formal derivations in T, and that T is complete means that for every (closed) formula A in the language of T, either A or its negation ~A has a formal derivation in T.

Thus, Gödel's theorem in the form "there is no T satisfying [certain conditions] which is both consistent and complete" doesn't involve the notions of truth or falsity. To apply the theorem to a theory T it isn't necessary to have any notion of truth or falsity for the formulas in the language of T. For example, we can note that axiomatic set theory is, by Gödel's theorem, incomplete or inconsistent, without assuming that the statements of formalized set theory are at all meaningful, or that it makes any sense to speak of them as true or false.

### Gödel's theorem as relating to truth

However, for the mathematical theories T most often discussed in connection with Gödel's theorem, such as formal arithmetic or various set theories we do have an interpretation of the sentences of the theory, and we do speak of formulas of the theory as true or false. In such cases we can conclude from the proof of Gödel's theorem e.g. that the Gödel sentence G is true (on its ordinary interpretation) if T is consistent.

Thus comments like the following are not justified:

So, again, in sum, all this chat of Platonism, truth and what not, is irrelevant to Goedel's incompleteness theorem. To try to impose such interpretations on this theorem is plain violence.
That the Gödel sentence for e.g. formalized arithmetic PA is true is a mathematical consequence of the consistency of PA. Here "the Gödel sentence is true" doesn't have any special philosophical meaning, but is just ordinary mathematical language. For example, to say that Goldbach's conjecture is true is to say that every even number greater than two is the sum of two primes, and similarly with the Gödel sentence.

In fact, Gödel himself very much thought in terms of truth in arriving at his theorem. He wrote in a letter (1966):

I think the theorem of mine which von Neumann refers to is not that on the existence of undecidable propositions or that on the length of proofs, but rather the fact that a complete epistemological description of a language A cannot be given in the same language A, because the concept of truth of sentences of A cannot be defined in A. It is this theorem which is the true reason for the existence of undecidable propositions in the formal systems containing arithmetic.
What Gödel is here referring to is what is commonly called Tarski's theorem, which states that the concept "true arithmetical sentence" is not definable in arithmetical terms. This was in fact first discovered by Gödel, when he found that self-referential statements could be formulated in arithmetic. So if "true arithmetical sentence" could be defined in arithmetic, the paradoxical statement "This statement is not true" could also be formulated in arithmetic, leading to a contradiction. On the other hand, Gödel found, the concept "arithmetical sentence provable in T" could be formulated in arithmetic itself. It follows that arithmetical truth cannot be the same as provability in T. In his 1931 proof of the incompleteness theorem, Gödel did not bring forward any of these considerations, because he did not wish to emphasize the philosophically more controversial concept of truth, but rather the purely formal one of provability.