# Conservative extension

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**.

## Examples[edit]

- , 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 .
^{[1]} - The subsystem is a -conservative extension of , and a -conservative over (primitive recursive arithmetic).
^{[1]} - 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 ().
- Extensions by definitions are conservative.
- Extensions by unconstrained predicate or function symbols are conservative.
- (a subsystem of Peano arithmetic with induction only for -formulas) is a -conservative extension of .
^{[2]} - is a -conservative extension of by Shoenfield's absoluteness theorem.
- with the continuum hypothesis is a -conservative extension of .
^{[citation needed]}

## Model-theoretic conservative extension[edit]

With model-theoretic means, a stronger notion is obtained: an extension of a theory is **model-theoretically conservative** if and every model of can be expanded to a model of . Each model-theoretic conservative extension also is a (proof-theoretic) conservative extension in the above sense.^{[3]} 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.

## See also[edit]

## References[edit]

- ^
^{a}^{b}S. G. Simpson, R. L. Smith, "Factorization of polynomials and -induction" (1986). Annals of Pure and Applied Logic, vol. 31 (p.305) **^**Fernando Ferreira, A Simple Proof of Parsons' Theorem. Notre Dame Journal of Formal Logic, Vol.46, No.1, 2005.**^**Hodges, Wilfrid (1997).*A shorter model theory*. Cambridge: Cambridge University Press. p. 58 exercise 8. ISBN 978-0-521-58713-6.