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, or proper extension, is a supertheory which is not conservative, and can prove more theorems than the original.
More formally stated, a theory is a 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 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.

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 .
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.
is a -conservative extension of.
is a [analytical hierarchy|]-conservative extension of by Shoenfield's absoluteness theorem.
with the generalized 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 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.