Slutsky equation

In mathematical logic, a logical theory $T_{2}$ is a (proof theoretic) conservative extension of a theory $T_{1}$ if the language of $T_{2}$ extends the language of $T_{1}$; every theorem of $T_{1}$ is a theorem of $T_{2}$; and any theorem of $T_{2}$ that is in the language of $T_{1}$ is already a theorem of $T_{1}$.

More generally, if Γ is a set of formulas in the common language of $T_{1}$ and $T_{2}$, then $T_{2}$ is Γ-conservative over $T_{1}$ if every formula from Γ provable in $T_{2}$ is also provable in $T_{1}$.

To put it informally, the new theory may possibly be more convenient for proving theorems, but it proves no new theorems about the language of the old theory.

Note that a conservative extension of a consistent theory is 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.

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).
• 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.

External links

• The importance of conservative extensions for the foundations of mathematics