[FOM] Re: Godel's Theorems

[Matt Insall]

Harvey Friedman asked:
>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?


I cannot speak for others, or to the question of overall agreement, but I
agree with it.

Harvey continues:
>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.

My difficulty with 4 is the phrase ``would be''.  To me, it WAS
a major event in the philosophy of mathematics, although I do not
think of it as being as surprising as I think many authors seem to
think it is.  That is, when I learned about *) and the meaning it
has, when correctly stated, I found it quite reasonable to not expect
PA to be able to prove its own consistency, and therefore the statement
and proof of *) are not so surprising.  (I do not mean to suggest that
it is not interesting, however.)


Matt Insall