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, or proper extension,[citation needed] is a supertheory which is not conservative, and can prove more theorems than the original.
More formally stated, a theory T2 is a (proof theoretic) conservative extension of a theory T1 if every theorem of T1 is a theorem of T2, and any theorem of T2 in the language of T1 is already a theorem of T1.
More generally, if Γ is a set of formulas in the common language of T1 and T2, then T2 is Γ-conservative over T1 if every formula from Γ provable in T2 is also provable in T1.
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 T2 would be a theorem of T2, so every formula in the language of T1 would be a theorem of T1, so T1 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, T0, that is known (or assumed) to be consistent, and successively build conservative extensions T1, T2, … of it.
Recently, conservative extensions have been used for defining a notion of module for ontologies[citation needed]: 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.
Examples
[edit]- ACA0, 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 RCA∗
0 and WKL∗
0 are Π0
2-conservative over EFA.[1] - The subsystem WKL0 is a Π1
1-conservative extension of RCA0, and a Π0
2-conservative over PRA.[1] - NBG is a conservative extension of ZFC (= ZF+AC).
- Internal set theory is a conservative extension of 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 Σ0
1-formulas) is a Π0
2-conservative extension of PRA.[2] - ZFC is a Π1
4-conservative extension of ZF by Shoenfield's absoluteness theorem.[3] - ZFC with the generalized continuum hypothesis is a Π2
1-conservative extension of ZFC.[4]
Model-theoretic conservative extension
[edit]With model-theoretic means, a stronger notion is obtained: an extension T2 of a theory T1 is model-theoretically conservative if T1 ⊆ T2 and every model of T1 can be expanded to a model of T2. Each model-theoretic conservative extension also is a (proof-theoretic) conservative extension in the above sense.[5] 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 Σ0
1-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.
- ^ Michael Rathjen, Power Kripke-Platek set theory and the axiom of choice. Journal of Logic and Computation, Vol.30, No.1, 2018.
- ^ Richard Platek, Eliminating the continuum hypothesis. The Journal of Symbolic Logic, Vol.36, No.1, 1969.
- ^ Hodges, Wilfrid (1997). A shorter model theory. Cambridge: Cambridge University Press. p. 58 exercise 8. ISBN 978-0-521-58713-6.
External links
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