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.