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 $\mathrm{\Gamma }$ is a set of formulas in the common language of $T_1$ and $T_2$, then $T_2$ is $\mathrm{\Gamma }$-conservative over $T_1$ if every formula from $\mathrm{\Gamma }$ 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

• ${\mathsf{A}\mathsf{C}\mathsf{A}}_0$, 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 ${\mathsf{R}\mathsf{C}\mathsf{A}}_0^{*}$ and ${\mathsf{W}\mathsf{K}\mathsf{L}}_0^{*}$ are ${\mathrm{\Pi }}_2^0$-conservative over $\mathsf{E}\mathsf{F}\mathsf{A}$.
• The subsystem ${\mathsf{W}\mathsf{K}\mathsf{L}}_0$ is a ${\mathrm{\Pi }}_1^1$-conservative extension of ${\mathsf{R}\mathsf{C}\mathsf{A}}_0$, and a ${\mathrm{\Pi }}_2^0$-conservative over $\mathsf{P}\mathsf{R}\mathsf{A}$ (primitive recursive arithmetic).
• Von Neumann–Bernays–Gödel set theory ($\mathsf{N}\mathsf{B}\mathsf{G}$) is a conservative extension of Zermelo–Fraenkel set theory with the axiom of choice ($\mathsf{Z}\mathsf{F}\mathsf{C}$).
• Internal set theory is a conservative extension of Zermelo–Fraenkel set theory with the axiom of choice ($\mathsf{Z}\mathsf{F}\mathsf{C}$).
• Extensions by definitions are conservative.
• Extensions by unconstrained predicate or function symbols are conservative.
• $I{\mathrm{\Sigma }}_1$ (a subsystem of Peano arithmetic with induction only for ${\mathrm{\Sigma }}^{0}1$-formulas) is a ${\mathrm{\Pi }}_2^0$-conservative extension of $\mathsf{P}\mathsf{R}\mathsf{A}$.
• $\mathsf{Z}\mathsf{F}\mathsf{C}$ is a ${\mathrm{\Sigma }}^{1}3$-conservative extension of $\mathsf{Z}\mathsf{F}$ by Shoenfield's absoluteness theorem.
• $\mathsf{Z}\mathsf{F}\mathsf{C}$ with the continuum hypothesis is a ${\mathrm{\Pi }}_1^2$-conservative extension of $\mathsf{Z}\mathsf{F}\mathsf{C}$.

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