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 ${\displaystyle T_{2))$ 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^{[citation needed]}: 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.

• ${\displaystyle {\mathsf {ACA))_{0))$, 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 ${\displaystyle \Sigma _{3}^{1))$-conservative extension of ${\displaystyle {\mathsf {ZF))}$ by Shoenfield's absoluteness theorem.
• ${\displaystyle {\mathsf {ZFC))}$ with the continuum hypothesis is a ${\displaystyle \Pi _{1}^{2))$-conservative extension of ${\displaystyle {\mathsf {ZFC))}$.^{[citation needed]}

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.^[3] 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.

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

• The importance of conservative extensions for the foundations of mathematics

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