Harman Patil (Editor)

Conservative extension

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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.

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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 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 .

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).
  • 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

    Conservative extension Wikipedia
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