# Conservative extension

In mathematical logic, a theory is a (proof theoretic) **conservative extension** of a theory if the language of extends the language of ; every theorem of is a theorem of ; and any theorem of that is in the language of is already a theorem of .

More generally, if Γ is a set of formulas in the common language of and , then is **Γ-conservative** over if every formula from Γ provable in is also provable in .

To put it informally, the new theory may possibly be more convenient for proving theorems, but it proves no new theorems about the language of the old theory.

Note that a conservative extension of a consistent theory is consistent. [If it were not, then by the principle of explosion ("everything follows from a contradiction"), every theorem in the original theory *as well as its negation* would belong to the new theory, which then would not be a conservative extension.] 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, , that is known (or assumed) to be consistent, and successively build conservative extensions , , ... of it.

The theorem provers Isabelle and ACL2 adopt this methodology by providing a language for conservative extensions by definition.

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

- ACA
_{0}(a subsystem of second-order arithmetic) is a conservative extension of first-order Peano arithmetic. - Von Neumann–Bernays–Gödel set theory is a conservative extension of Zermelo–Fraenkel set theory with the axiom of choice (ZFC).
- Internal set theory is a conservative extension of Zermelo–Fraenkel set theory with the axiom of choice (ZFC).
- Extensions by definitions are conservative.
- Extensions by unconstrained predicate or function symbols are conservative.
- IΣ
_{1}(a subsystem of Peano arithmetic with induction only for Σ^{0}_{1}-formulas) is a Π^{0}_{2}-conservative extension of the primitive recursive arithmetic (PRA).^{[1]} - ZFC is a Π
^{1}_{3}-conservative extension of ZF by Shoenfield's absoluteness theorem. - ZFC with the continuum hypothesis is a Π
^{2}_{1}-conservative extension of ZFC.

## Model-theoretic conservative extension

With model-theoretic means, a stronger notion is obtained: an extension of a theory is **model-theoretically conservative** if every model of can be expanded to a model of . It is straightforward to see that 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.