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
Recently, conservative extensions have been used for defining a notion of
An extension which is not conservative may be called a proper extension.
Examples
- , a subsystem of 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
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
References
- ^ 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.
- ISBN 978-0-521-58713-6.