Conservative extension

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In mathematical logic, a logical theory $T 2$ 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.

[edit] 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.

[edit] 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.

See also: Conservativity theorem