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 ${\displaystyle T_{2}}$ is a (proof theoretic) conservative extension of a theory ${\displaystyle T_{1}}$ if every theorem of ${\displaystyle T_{1}}$ is a theorem of ${\displaystyle T_{2}}$, and any theorem of ${\displaystyle T_{2}}$ in the language of ${\displaystyle T_{1}}$ is already a theorem of ${\displaystyle T_{1}}$.

More generally, if ${\displaystyle \Gamma }$ is a set of formulas in the common language of ${\displaystyle T_{1}}$ and ${\displaystyle T_{2}}$, then ${\displaystyle T_{2}}$ is ${\displaystyle \Gamma }$-conservative over ${\displaystyle T_{1}}$ if every formula from ${\displaystyle \Gamma }$ provable in ${\displaystyle T_{2}}$ is also provable in ${\displaystyle T_{1}}$.

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 ${\displaystyle T_{2}}$ would be a theorem of ${\displaystyle T_{2}}$, so every formula in the language of ${\displaystyle T_{1}}$ would be a theorem of ${\displaystyle T_{1}}$, so ${\displaystyle T_{1}}$ 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, ${\displaystyle T_{0}}$, that is known (or assumed) to be consistent, and successively build conservative extensions ${\displaystyle T_{1}}$, ${\displaystyle T_{2}}$, ... 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.

An extension which is not conservative may be called a proper extension.

Examples

• ${\displaystyle {\mathsf {ACA}}_{0}}$, 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 ${\displaystyle {\mathsf {RCA}}_{0}^{*}}$ and ${\displaystyle {\mathsf {WKL}}_{0}^{*}}$ are ${\displaystyle \Pi _{2}^{0}}$-conservative over ${\displaystyle {\mathsf {EFA}}}$. ^[1]
• The subsystem ${\displaystyle {\mathsf {WKL}}_{0}}$ is a ${\displaystyle \Pi _{1}^{1}}$-conservative extension of ${\displaystyle {\mathsf {RCA}}_{0}}$, and a ${\displaystyle \Pi _{2}^{0}}$-conservative over ${\displaystyle {\mathsf {PRA}}}$ (primitive recursive arithmetic). ^[1]
• Von Neumann–Bernays–Gödel set theory (${\displaystyle {\mathsf {NBG}}}$) is a conservative extension of Zermelo–Fraenkel set theory with the axiom of choice (${\displaystyle {\mathsf {ZFC}}}$).
• Internal set theory is a conservative extension of Zermelo–Fraenkel set theory with the axiom of choice (${\displaystyle {\mathsf {ZFC}}}$).
• Extensions by definitions are conservative.
• Extensions by unconstrained predicate or function symbols are conservative.
• ${\displaystyle I\Sigma _{1}}$ (a subsystem of Peano arithmetic with induction only for ${\displaystyle \Sigma _{1}^{0}}$-formulas) is a ${\displaystyle \Pi _{2}^{0}}$-conservative extension of ${\displaystyle {\mathsf {PRA}}}$. ^[2]
• ${\displaystyle {\mathsf {ZFC}}}$ is a ${\displaystyle \Sigma _{3}^{1}}$-conservative extension of ${\displaystyle {\mathsf {ZF}}}$ by Shoenfield's absoluteness theorem.
• ${\displaystyle {\mathsf {ZFC}}}$ with the continuum hypothesis is a ${\displaystyle \Pi _{1}^{2}}$-conservative extension of ${\displaystyle {\mathsf {ZFC}}}$.^{[ citation needed ]}

Model-theoretic conservative extension

With model-theoretic means, a stronger notion is obtained: an extension ${\displaystyle T_{2}}$ of a theory ${\displaystyle T_{1}}$ is model-theoretically conservative if ${\displaystyle T_{1}\subseteq T_{2}}$ and every model of ${\displaystyle T_{1}}$ can be expanded to a model of ${\displaystyle T_{2}}$. 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

• Extension by definitions
• Extension by new constant and function names

References

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

External links

• The importance of conservative extensions for the foundations of mathematics