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 $T_{2)$ is a (proof theoretic) conservative extension of a theory $T_{1)$ if every theorem of $T_{1)$ is a theorem of $T_{2)$ , and any theorem of $T_{2)$ in the language of $T_{1)$ is already a theorem of $T_{1)$ .

More generally, if $\Gamma$ is a set of formulas in the common language of $T_{1)$ and $T_{2)$ , then $T_{2)$ is $\Gamma$ -conservative over $T_{1)$ if every formula from $\Gamma$ provable in $T_{2)$ is also provable in $T_{1)$ .

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 $T_{2)$ would be a theorem of $T_{2)$ , so every formula in the language of $T_{1)$ would be a theorem of $T_{1)$ , so $T_{1)$ 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, $T_{0)$ , that is known (or assumed) to be consistent, and successively build conservative extensions $T_{1)$ , $T_{2)$ , ... 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

• ${\mathsf {ACA))_{0)$ , 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 ${\mathsf {RCA))_{0}^{*)$ and ${\mathsf {WKL))_{0}^{*)$ are $\Pi _{2}^{0)$ -conservative over ${\mathsf {EFA))$ .
• The subsystem ${\mathsf {WKL))_{0)$ is a $\Pi _{1}^{1)$ -conservative extension of ${\mathsf {RCA))_{0)$ , and a $\Pi _{2}^{0)$ -conservative over ${\mathsf {PRA))$ (primitive recursive arithmetic).
• Von Neumann–Bernays–Gödel set theory (${\mathsf {NBG))$ ) is a conservative extension of Zermelo–Fraenkel set theory with the axiom of choice (${\mathsf {ZFC))$ ).
• Internal set theory is a conservative extension of Zermelo–Fraenkel set theory with the axiom of choice (${\mathsf {ZFC))$ ).
• Extensions by definitions are conservative.
• Extensions by unconstrained predicate or function symbols are conservative.
• $I\Sigma _{1)$ (a subsystem of Peano arithmetic with induction only for $\Sigma _{1}^{0)$ -formulas) is a $\Pi _{2}^{0)$ -conservative extension of ${\mathsf {PRA))$ .
• ${\mathsf {ZFC))$ is a $\Sigma _{3}^{1)$ -conservative extension of ${\mathsf {ZF))$ by Shoenfield's absoluteness theorem.
• ${\mathsf {ZFC))$ with the continuum hypothesis is a $\Pi _{1}^{2)$ -conservative extension of ${\mathsf {ZFC))$ .[citation needed]

## Model-theoretic conservative extension

With model-theoretic means, a stronger notion is obtained: an extension $T_{2)$ of a theory $T_{1)$ is model-theoretically conservative if $T_{1}\subseteq T_{2)$ and every model of $T_{1)$ can be expanded to a model of $T_{2)$ . 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.

1. ^ a b S. G. Simpson, R. L. Smith, "Factorization of polynomials and $\Sigma _{1}^{0)$ -induction" (1986). Annals of Pure and Applied Logic, vol. 31 (p.305)