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

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

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

• ${\displaystyle {\mathsf {ACA}}_{0}}$, 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 ${\displaystyle {\mathsf {RCA}}_{0}^{*}}$ and ${\displaystyle {\mathsf {WKL}}_{0}^{*}}$ are ${\displaystyle \Pi _{2}^{0}}$-conservative over ${\displaystyle {\mathsf {EFA}}}$.¹
• The subsystem ${\displaystyle {\mathsf {WKL}}_{0}}$ is a ${\displaystyle \Pi _{1}^{1}}$-conservative extension of ${\displaystyle {\mathsf {RCA}}_{0}}$, and a ${\displaystyle \Pi _{2}^{0}}$-conservative over ${\displaystyle {\mathsf {PRA}}}$ (primitive recursive arithmetic).¹
• Von Neumann–Bernays–Gödel set theory (${\displaystyle {\mathsf {NBG}}}$) is a conservative extension of Zermelo–Fraenkel set theory with the axiom of choice (${\displaystyle {\mathsf {ZFC}}}$).
• Internal set theory is a conservative extension of Zermelo–Fraenkel set theory with the axiom of choice (${\displaystyle {\mathsf {ZFC}}}$).
• Extensions by definitions are conservative.
• Extensions by unconstrained predicate or function symbols are conservative.
• ${\displaystyle I\Sigma _{1}}$ (a subsystem of Peano arithmetic with induction only for ${\displaystyle \Sigma _{1}^{0}}$-formulas) is a ${\displaystyle \Pi _{2}^{0}}$-conservative extension of ${\displaystyle {\mathsf {PRA}}}$.²
• ${\displaystyle {\mathsf {ZFC}}}$ is a ${\displaystyle \Pi _{4}^{1}}$-conservative extension of ${\displaystyle {\mathsf {ZF}}}$ by Shoenfield's absoluteness theorem.³
• ${\displaystyle {\mathsf {ZFC}}}$ with the generalized continuum hypothesis is a ${\displaystyle \Pi _{1}^{2}}$-conservative extension of ${\displaystyle {\mathsf {ZFC}}}$.⁴

With model-theoretic means, a stronger notion is obtained: an extension ${\displaystyle T_{2}}$ of a theory ${\displaystyle T_{1}}$ is model-theoretically conservative if ${\displaystyle T_{1}\subseteq T_{2}}$ and every model of ${\displaystyle T_{1}}$ can be expanded to a model of ${\displaystyle T_{2}}$. Each model-theoretic conservative extension also is a (proof-theoretic) conservative extension in the above sense.⁵ 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.