Is it true that a conservative extension of a theory is equiconsistent with it, and, if so, why?
WP says: "In mathematical logic, a logical theory T2 is a (proof theoretic) conservative extension of a theory T1 if the language of T2 extends the language of T1; every theorem of T1 is a theorem of T2; and any theorem of T2 which is in the language of T1 is already a theorem of T1. [...] Note that a conservative extension of a consistent theory is consistent."
I have a copy of Hodges, A shorter model theory, which gives a definition on p. 58 that seems to be exactly equivalent, although the claim about equiconsistency isn't made.
So what's wrong with the following counterexample?
T1 is a theory in which the only sentence is A, and the only axiom is that A is true.
T2 is a theory in which the only sentences are A and B. It has three axioms: (1) A is true. (2) B is true. (3) B is false.
T2 is a conservative extension of T1, but although T1 is consistent, T2 is inconsistent.