[pam69ur]

The hypotheses of science will be forever unproven. That is, the past success of a scientific hypothesis does not imply its future success, and so by convention scientists hold every scientific hypothesis open to refutation. However, the elementary fact that scientific hypothesis are unproven is not generally considered a cause for alarm, and neither is it thought a damning criticism. In other words, the unprovable status of scientific hypotheses does not imply that they are false, and since scientific investigation is concerned with discovering true theories, and not provably true theories, the unprovability of scientific theories is not something to be too worried about.

In a like manner, there are statements about some formal languages which cannot be proven to be true by that language--else the language must be inconsistent. Therefore, if a language is consistent then it is impossible to construct a proof for every statement of that language, and so mathematicians, like scientists, by convention hold their hypotheses open to refutation. However, that it is impossible to prove, once and for all, that such a language is consistent is not a cause for alarm, and neither is it a damning criticism. In other words, the unprovable status of some mathematical languages does not imply that they are inconsistent, and since it is their consistency which we are concerned with, that they might be unprovable is not something to be too worried about.

all irrelevant

as colin ****** dean has noted

ZFC cannot prove anything as it has statements which cant be proven

in ZFC if there are statements which canot be proven -like the once Taniyama–Shimura conjecture- then ZFC cant be used to prove anything but only give conjectures


if it applies to wiles proof
it then applies to ZFC
ie with out proof of all the statements -ieTaniyama–Shimura conjecture- then ZFC cant proove anything

if wiles gave a proof using an unproven Taniyama–Shimura conjecture-mathematicians would have said he did not give a proof as one of his statements was itself unproven

so
mathematicans must then say ZFC cant give any proof as some of its statements are unproven
VERY SIMPLE
what applies to wiles must then apply to ZFC