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, every formula in the language of would be a theorem of , so every formula in the language of would be a theorem of , so 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, , 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.
Examples
, 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 and are -conservative over . The subsystem is a -conservative extension of , and a -conservative over (primitive recursive arithmetic). Von Neumann–Bernays–Gödel set theory () is a conservative extension of Zermelo–Fraenkel set theory with the axiom of choice (). Internal set theory is a conservative extension of Zermelo–Fraenkel set theory with the axiom of choice (). Extensions by definitions are conservative. Extensions by unconstrained predicate or function symbols are conservative. (a subsystem of Peano arithmetic with induction only for -formulas) is a -conservative extension of . is a -conservative extension of by Shoenfield's absoluteness theorem. with the continuum hypothesis is a -conservative extension of .
Model-theoretic conservative extension
With model-theoretic means, a stronger notion is obtained: an extension of a theory is model-theoretically conservative if and every model of can be expanded to a model of . 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.