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[citation needed]: 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
.[1]
- The subsystem
is a
-conservative extension of
, and a
-conservative over
(primitive recursive arithmetic).[1]
- 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
.[2]
is a
-conservative extension of
by Shoenfield's absoluteness theorem.
with the continuum hypothesis is a
-conservative extension of
.[citation needed]
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.[3] 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.