If I have a set of axioms A = {A1, ..., An} and if I create a set of axioms B = {A1, ..., An, Con(A)}, would it be true to say that Con(A) iff Con(B)? Is there a simple counter-example to this?
More generally, if something X is true of A and we construct C = {A1, ..., An, X}, does Con(A) bi-imply Con(C)? Thanks.