Conservative extension

In mathematical logic, a conservative extension is a subtheory of a theory which is often convenient for proving theorems, but proves no new theorems about the language of the original theory.

More formally stated, a theory $T_2$ is a (proof theoretic) conservative extension of a theory $T_1$ if the language of $T_2$ extends the language of $T_1$; that is, 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 ("everything follows from a contradiction"), every theorem in the original theory as well as its negation would belong to the new theory, which then would not be a conservative extension.] 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.

The theorem provers Isabelle and ACL2 adopt this methodology by providing a language for conservative extensions by definition.

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

• ACA₀ (a subsystem of second-order arithmetic) is a conservative extension of first-order Peano arithmetic.
• Von Neumann–Bernays–Gödel set theory is a conservative extension of Zermelo–Fraenkel set theory with the axiom of choice (ZFC).
• Internal set theory is a conservative extension of Zermelo–Fraenkel set theory with the axiom of choice (ZFC).
• Extensions by definitions are conservative.
• Extensions by unconstrained predicate or function symbols are conservative.
• IΣ₁ (a subsystem of Peano arithmetic with induction only for Σ⁰₁-formulas) is a Π⁰₂-conservative extension of the primitive recursive arithmetic (PRA).^[1]
• ZFC is a Π¹₃-conservative extension of ZF by Shoenfield's absoluteness theorem.
• ZFC with the continuum hypothesis is a Π²₁-conservative extension of 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 every model of $T_1$ can be expanded to a model of $T_2$. It is straightforward to see that 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.

References

1. Notre Dame Journal of Formal Logic, Fernando Ferreira, A simple proof of Parsons’ theorem

External links

• The importance of conservative extensions for the foundations of mathematics