Extension conservatrice

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 is a (proof theoretic) conservative extension of a theory if every theorem of is a theorem of , and any theorem of in the language of is already a theorem of . More generally, if is a set of formulas in the common language of and , then is -conservative over if every formula from provable in is also provable in . 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 would be a theorem of , so every formula in the language of would be a theorem of , so 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, , that is known (or assumed) to be consistent, and successively build conservative extensions , , ... 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. 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 and are -conservative over . The subsystem is a -conservative extension of , and a -conservative over (primitive recursive arithmetic). Von Neumann–Bernays–Gödel set theory () is a conservative extension of Zermelo–Fraenkel set theory with the axiom of choice (). Internal set theory is a conservative extension of Zermelo–Fraenkel set theory with the axiom of choice ().