Conservative extension of an empty theory

Question

Say we have the language $L = \{C, P\}$ where $C$ is a nullary predicate and $P$ a unary one. Then we consider the empty theory $E$ over the language $L$. What does a conservative extension of $E$ look like? My intuition is that it can contain no formulas with the $C, P$ predicates, is that correct?

Answer 1

You're essentially right, with a couple caveats:

• Not every sentence not using $C,P$ is conservative over $E$: for example, "$\forall x,y(x=y)$."

• There are sentences involving $C,P$ in trivial ways which are conservative over $E$: for example, "$\forall x(P(x)\vee \neg P(x))$."

It's also worth pointing out that there's nothing special about the specific language $L$ here. In general, if $T_0$ is a conservative extension of $T_1$ then $T_0$ won't have any axioms using the symbols of the language of $T_1$ in a "nontrivial" way which don't follow from $T_1$ already, but this is not a sufficient condition for conservativity.