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In mathematical logic, a theory is a (proof theoretic) conservative extension of a theory if the language of extends the language of ; every theorem of is a theorem of ; and any theorem of that is in the language of is already a theorem of . More generally, if Γ is a set of formulas in the common language of and , then is Γ-conservative over if every formula from Γ provable in is also provable in . 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. (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, , that is known (or assumed) to be consistent, and successively build conservative extensions , , ... 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).〔(Notre Dame Journal of Formal Logic, Fernando Ferreira, A simple proof of Parsons’ theorem )〕 * 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. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「conservative extension」の詳細全文を読む スポンサード リンク
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