What's the 'real' way of saying 'cautiously extends'? Its well known that ZF is equiconsistent with ZFC. Thus we say 'the Axiom of Choice cautiously extends ZF'.
Except we don't, because I just made that up. What's the usual way of saying this sort of thing?
 A: Note that the equiconsistency of a theory $T + \phi$ relative to $T$ says, for theories that include or interpret arithmetic, that if $T + \phi$ proves $0=1$ then $T$ proves $0=1$. 
We can broaden this notation in a certain way. Let $C$ be a set of formulas. We say that a theory $T'$ is conservative over a theory $T$ for formulas in $C$ if, for all $\phi \in C$, if $T'$ proves $C$ then $T$ proves $C$. In this case $T'$ is no stronger, with respect to formulas in $C$, than $T$ already was. If $T \subseteq T'$ we say $T'$ is a conservative extension for formulas in $C$. 
Equiconsistency is thus a very weak form of conservativity. It is common in practice to establish much stronger conservation results. For example, one consequence of Shoenfield's absoluteness theorem is that 

ZFC is conservative over ZF for all sentences in the language of Peano
  arithmetic (or, strictly speaking, the interpretations of these sentences into
  the the language of set theory). 

One such sentence is $0 =1$, but this conservation result is much stronger than that. Fermat's last theorem, the Riemann hypothesis, and other interesting results can be phrased as sentences of Peano arithmetic. Thus, if ZFC proves these, they are already provable in ZF. 
