[FOM] Godel's Theorems

[Harvey Friedman]

I looked at

On Floyd and Putnam on Wittgenstein on Godel,  http://www.nd.edu/~tbays/papers/

and

http://staff.washington.edu/dalexand/Putnam%20Readings/Notorious.pdf

I have some specific questions. Let PA be the usual Peano Arithmetic.
Consider the claim

*) there is a true sentence in the language of PA which is not provable in PA.

1. Conventional wisdom is that this is now a fully established
theorem of mathematics (or ordinary mathematics as currently
practiced by the overwhelming majority of mathematicians). Is there
agreement on this?

2. For those who do not agree, do they believe that *) is not a
mathematical statement capable of mathematical proof? E.g., this
could be on the grounds that they do not accept the usual
mathematical definition of "true sentence in the language of PA".

3. For those who believe that *) is a mathematical statement capable
of mathematical proof, but do not agree with 1. Do you see a flawed
step in the mathematical proof of *)? E.g., that it uses some
questionable inductions and/or definitions by recursion.

On a related but separate, matter,

4. Conventional wisdom is that the establishing of *) would be a very
major event in philosophy of mathematics and/or foundations of
mathematics. Do you agree with this?

5. For those who do not agree with 4, please elaborate.