peano arithmetic - How would one even begin to try to prove that a simple number-theoretic statement is undecidable?

EDIT

Here is my problem. To prove that statement S is undecidable is to

(1) prove that one cannot prove S.

I think I understand the meaning of the second "prove". (It depends of course on the context.) But I don't understand the meaning of the first "prove".

A "solution" would be to replace (1) by

(2) give an informal proof showing that one cannot prove S,

or

(3) give an acceptable argument showing that one cannot prove S.

This seems to be in tune with the following quotation from Cohen's "Set theory and the Continuum Hypothesis" (p. 41):

We have now arrived at a rather peculiar situation. On the one hand $sim A$ is not provable in $Z_1$ and yet we have just given an informal proof that $sim A$ is true. (There is no contradiction here since we have merely shown that the proofs in $Z_1$ do not exhaust the set of all acceptable arguments.)

I think it would be an enormous progress to replace (1) by (2) or (3).

In other words, instead of talking about "proving" that some statements are undecidable, it would be wiser, I believe, to talk about giving "informal proofs", or "acceptable arguments", or "convincing evidence", ... that these statements are undecidable.

END OF EDIT

---

Here is, for what it's worth, my personal conviction on this.

If (like me) you don't believe that "you can't get something for nothing", then you don't believe in Gödel and Cohen's results.

The claim to "get something for nothing" is very openly expressed (in my opinion) by Cohen on page 39 of "Set theory and the Continuum Hypothesis":

The theorems of the previous section are not results about what can be proved in particular axiom systems; they are absolute statements about functions.

Cohen really says: "The theorems of the previous section are proved without invoking any axiom, that is, they are gotten for nothing". Or am I putting words in his mouth?

I think the key is to understand the respective STATUS of the various statements involved. In particular, a clear distinction should be made between mathematical and metamathematical statements.

I also think we should all make an effort to talk unemotionally about such questions.