Therefore, it is not necessary that your second premise is fulfilled before ZFC can be used in proof construction, since ZFCs consistency is a property independent of what we can or cannot prove about ZFC or anything else

[1] In this regard the law of identity is the ultimate foundation upon which logic rests, without an 'identity' (for the symbols of logic) logic is overthrown and collapses.......The law of non-contradiction to quote Aristotle states " the same attribute (characteristic essence) cannot at the same time belong and not belong to the same subject and in the same respect."[3] In terms of propositional calculus ' it is not the case both p and not p'. In this regard we see that if there is no essence to characterise a subject in distinction from other subjects there can be no law of non-contradiction and thus no logic at all. In other words if there is nothing to distinguish a 'horse' from a 'non-horse', either ontological or nominal, in the proposition P 'there is a horse' then we can not apply the law of non-contradiction because we have no distinguishable subject for the subject of the proposition.

[1] W. R. B, Gibson, 1908, p,95.

[2] C, Dean, op. cit. p. XXV-XXXV.

[3] A, Flew, 1979, p.75.

In mathematics, a conjecture is a mathematical statement which appears likely to be true, but has not been formally proven to be true under the rules of mathematical logic

In any case, my original point was the validity of an inference does not depend on our ability to prove that it is valid, just as the truth of a proposition does not depend on our ability to prove that it is true i.e. if a proposition true, then it will be true before we have proven that it is true, after we have proven that it is true, and even if we cannot prove that it is true. For example, if I use '1 + 1 = 2' in a mathematical proof, then it is expected that I can or will also develop a proof for '1 + 1 = 2', but the validity of my argument does not depend on whether I have proven '1 + 1 = 2', but only on whether '1 + 1 = 2' is a valid inference. In other words, the validity of an argument depends upon whether or not the rules of inference used are truth-preserving, and not on whether those rules have been proven to be truth-preserving.

In the summer of 1986, Ken Ribet succeeded in proving the epsilon conjecture. (His article was published in 1990.) He demonstrated that, just as Frey had anticipated, a special case of the Taniyama-Shimura conjecture (still unproven at the time), together with the now proven epsilon conjecture, implies Fermat's Last Theorem. Thus, if the Taniyama-Shimura conjecture holds for a class of elliptic curves called semistable elliptic curves, then Fermat's Last Theorem would be true.
After learning about Ribet's work, Andrew Wiles set out to prove that every semistable elliptic curve is modular. He did so in almost complete secrecy, working for a full seven years with minimal outside help. Over the course of three lectures delivered at Isaac Newton Institute for Mathematical Sciences on June 21, 22, and 23 of 1993, Wiles announced his proof of the Taniyama-Shimura conjecture, and hence of the Fermat's Last Theorem. Wiles drew upon a wide variety of methods in the proof, some of them having been developed especially for this occasion.

That a statement or argument is conjectural does not imply that it is false or invalid, and it is not a criticism of such a statement of argument that it might be false or invalid.

If you actually have a criticism of ZFC, rather than the trivial fact that it could possibly be mistaken, please provide it.

same with 1+1=2 untill it is proven all you can do with it is giive conjectures

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