Conservative extension

In mathematical logic, a conservative extension is a supertheory of a theory which is often convenient for proving theorems, but proves no new theorems about the language of the original theory. Similarly, a non-conservative extension is a supertheory which is not conservative, and can prove more theorems than the original.

More formally stated, a theory $$T_2$$ is a (proof theoretic) conservative extension of a theory $$T_1$$ if every theorem of $$T_1$$ is a theorem of $$T_2$$, and any theorem of $$T_2$$ in the language of $$T_1$$ is already a theorem of $$T_1$$.

More generally, if $$\Gamma$$ is a set of formulas in the common language of $$T_1$$ and $$T_2$$, then $$T_2$$ is $$\Gamma$$-conservative over $$T_1$$ if every formula from $$\Gamma$$ provable in $$T_2$$ is also provable in $$T_1$$.

Note that a conservative extension of a consistent theory is consistent. If it were not, then by the principle of explosion, every formula in the language of $$T_2$$ would be a theorem of $$T_2$$, so every formula in the language of $$T_1$$ would be a theorem of $$T_1$$, so $$T_1$$ would not be consistent. Hence, conservative extensions do not bear the risk of introducing new inconsistencies. This can also be seen as a methodology for writing and structuring large theories: start with a theory, $$T_0$$, that is known (or assumed) to be consistent, and successively build conservative extensions $$T_1$$, $$T_2$$, ... of it.

Recently, conservative extensions have been used for defining a notion of module for ontologies: if an ontology is formalized as a logical theory, a subtheory is a module if the whole ontology is a conservative extension of the subtheory.

An extension which is not conservative may be called a proper extension.

Examples

 * $$\mathsf{ACA}_0$$, a subsystem of second-order arithmetic studied in reverse mathematics, is a conservative extension of first-order Peano arithmetic.
 * The subsystems of second-order arithmetic $$\mathsf{RCA}_0^*$$ and $$\mathsf{WKL}_0^*$$ are $$\Pi_2^0$$-conservative over $$\mathsf{EFA}$$.
 * The subsystem $$\mathsf{WKL}_0$$ is a $$\Pi_1^1$$-conservative extension of $$\mathsf{RCA}_0$$, and a $$\Pi_2^0$$-conservative over $$\mathsf{PRA}$$ (primitive recursive arithmetic).
 * Von Neumann–Bernays–Gödel set theory ($$\mathsf{NBG}$$) is a conservative extension of Zermelo–Fraenkel set theory with the axiom of choice ($$\mathsf{ZFC}$$).
 * Internal set theory is a conservative extension of Zermelo–Fraenkel set theory with the axiom of choice ($$\mathsf{ZFC}$$).
 * Extensions by definitions are conservative.
 * Extensions by unconstrained predicate or function symbols are conservative.
 * $$I\Sigma_1$$ (a subsystem of Peano arithmetic with induction only for $\Sigma^0_1$-formulas) is a $$\Pi^0_2$$-conservative extension of $$\mathsf{PRA}$$.
 * $$\mathsf{ZFC}$$ is a $\Sigma^1_3$-conservative extension of $$\mathsf{ZF}$$ by Shoenfield's absoluteness theorem.
 * $$\mathsf{ZFC}$$ with the continuum hypothesis is a $$\Pi^2_1$$-conservative extension of $$\mathsf{ZFC}$$.

Model-theoretic conservative extension
With model-theoretic means, a stronger notion is obtained: an extension $$T_2$$ of a theory $$T_1$$ is model-theoretically conservative if $$T_1 \subseteq T_2$$ and every model of $$T_1$$ can be expanded to a model of $$T_2$$. Each model-theoretic conservative extension also is a (proof-theoretic) conservative extension in the above sense. The model theoretic notion has the advantage over the proof theoretic one that it does not depend so much on the language at hand; on the other hand, it is usually harder to establish model theoretic conservativity.