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Conservative extension

T_2

is a (proof theoretic) conservative extension of a theory

T_1

if the language of

T_2

extends the language of

T_1

and every theorem of

T_1

is a theorem of

T_2

and any theorem of

T_2

which is in the language of

T_1

is already a theorem of

T_1

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

Note that a conservative extension of a consistent theory is 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, $T_0$ , that is known (or assumed) to be consistent, and successively build conservative extensions $T_1$ , $T_2$ , ... 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

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.
Internal set theory is a conservative extension of Zermelo-Fraenkel set theory with the Axiom of choice.
Extensions by definitions are conservative.
Extensions by predicate or function symbols that are recursively-defined by a set of formulas are conservative
(provided that the recursion scheme leads to a definition).
Extensions by unconstrained predicate or function symbols are conservative.
Extensions by predicate or function symbols that are axiomatized by a Horn theory are conservative.

Model-theoretic conservative extension

With model-theoretic means, a stronger notion is obtained:

T_2

is a model-theoretic conservative extension of

T_1

if every model of

T_1

can be expanded to a model of

T_2

. 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.