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

* $\mathsf_0$, 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 $\mathsf_0^*$ and $\mathsf_0^*$ are $\Pi_2^0$-conservative over $\mathsf$.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)
* The subsystem $\mathsf_0$ is a $\Pi_1^1$-conservative extension of $\mathsf_0$, and a $\Pi_2^0$-conservative over $\mathsf$ ( primitive recursive arithmetic).
* Von Neumann–Bernays–Gödel set theory ($\mathsf$) is a conservative extension of Zermelo–Fraenkel set theory with the axiom of choice ($\mathsf$).
* Internal set theory is a conservative extension of Zermelo–Fraenkel set theory with the axiom of choice ($\mathsf$).
* 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 $\Sigma^0_1$-formulas) is a $\Pi^0_2$-conservative extension of $\mathsf$.
* $\mathsf$ is a $\Sigma^1_3$-conservative extension of $\mathsf$ by Shoenfield's absoluteness theorem.
* $\mathsf$ with the continuum hypothesis is a $\Pi^2_1$-conservative extension of $\mathsf$.

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 new constant and function names

References

External links

The importance of conservative extensions for the foundations of mathematics

{{Mathematical logic

Mathematical logic
Model theory
Proof theory