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 is a (proof theoretic) conservative extension of a theory if every theorem of is a theorem of , and any theorem of in the language of is already a theorem of .
More generally, if is a set of formulas in the common language of and , then is -conservative over if every formula from provable in is also provable in .
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.
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.
- ACA0 (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.
- IΣ1 (a subsystem of Peano arithmetic with induction only for Σ01-formulas) is a Π02-conservative extension of the primitive recursive arithmetic (PRA).
- ZFC is a Π13-conservative extension of ZF by Shoenfield's absoluteness theorem.
- ZFC with the continuum hypothesis is a Π21-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.