FOM: large cardinals and P vs NP

[Stephen Cook]

Further to the question of whether large cardinals might help
settle the P vs NP question, could someone answer the following
question:

I suppose that a large cardinal axiom could be consistent with ZFC,
but in some sense still false.  But in this case, could it imply
a false statement of number theory?

More precisely, suppose C is a large cardinal axiom and A is a statement
of first-order number theory.   Suppose that C + ZFC is consistent,
and  C+ ZFC implies A.  Is A necessarily true?

Stephen Cook