We can avoid the issue of truth by simply proving the following statement:
There is no Turing machine which, given a statement S, returns TRUE if S is provable in PA, returns FALSE if NOT(S) is provable in PA, and returns either TRUE or FALSE for all S.
Philosophers can then argue about what truth means. As long as you believe that PA is
consistent true, and that every statement is either true or false, this shows that a Turing machine cannot capture truth. A formalist may not agree with this interpretation of the above statement, but they certainly will agree that this statement is meaningful.
UPDATE: See discussion below of whether what a person who thinks PA is consistent, but not true, might make of this answer.