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 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 ${\mathsf {ZFC}}$.

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.

See also

• Extension by definitions
• Extension by new constant and function names

External links

• The importance of conservative extensions for the foundations of mathematics