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 $T_2$ is a (proof theoretic) conservative extension of a theory $T_1$ if every theorem of $T_1$ is a theorem of $T_2$ , and any theorem of $T_2$ in the language of $T_1$ is already a theorem of $T_1$ .

More generally, if $\Gamma$ is a set of formulas in the common language of $T_1$ and $T_2$ , then $T_2$ is $\Gamma$ -conservative over $T_1$ if every formula from $\Gamma$ provable in $T_2$ is also provable in $T_1$ .

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 $T_2$ would be a theorem of $T_2$ , so every formula in the language of $T_1$ would be a theorem of $T_1$ , so $T_1$ 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, $T_0$ , that is known (or assumed) to be consistent, and successively build conservative extensions $T_1$ , $T_2$ , ... of it.

Recently, conservative extensions have been used for defining a notion of module for ontologies{{citation needed|date=April 2024}}: 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

$\mathsf{ACA}_0$ , 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 $\mathsf{RCA}_0^*$ and $\mathsf{WKL}_0^*$ are $\Pi_2^0$ -conservative over $\mathsf{EFA}$ .S. G. Simpson, R. L. Smith, "[https://www.sciencedirect.com/science/article/pii/0168007286900746 Factorization of polynomials and $\Sigma_1^0$ -induction]" (1986). Annals of Pure and Applied Logic, vol. 31 (p.305)
The subsystem $\mathsf{WKL}_0$ is a $\Pi_1^1$ -conservative extension of $\mathsf{RCA}_0$ , and a $\Pi_2^0$ -conservative over $\mathsf{PRA}$ (primitive recursive arithmetic).
Von Neumann–Bernays–Gödel set theory ( $\mathsf{NBG}$ ) is a conservative extension of Zermelo–Fraenkel set theory with the axiom of choice ( $\mathsf{ZFC}$ ).
Internal set theory is a conservative extension of Zermelo–Fraenkel set theory with the axiom of choice ( $\mathsf{ZFC}$ ).
Extensions by definitions are conservative.
Extensions by unconstrained predicate or function symbols are conservative.
$I\Sigma_1$ (a subsystem of Peano arithmetic with induction only for $\Sigma^0_1$ -formulas) is a $\Pi^0_2$ -conservative extension of $\mathsf{PRA}$ .[https://projecteuclid.org/download/pdfview_1/euclid.ndjfl/1107220675A Fernando Ferreira, A Simple Proof of Parsons' Theorem. Notre Dame Journal of Formal Logic, Vol.46, No.1, 2005.]
$\mathsf{ZFC}$ is a $\Sigma^1_3$ -conservative extension of $\mathsf{ZF}$ by Shoenfield's absoluteness theorem.
$\mathsf{ZFC}$ with the continuum hypothesis is a $\Pi^2_1$ -conservative extension of $\mathsf{ZFC}$ .{{Citation needed|date=March 2021}}

Model-theoretic conservative extension

With model-theoretic means, a stronger notion is obtained: an extension $T_2$ of a theory $T_1$ is model-theoretically conservative if $T_1 \subseteq T_2$ and every model of $T_1$ can be expanded to a model of $T_2$ . Each model-theoretic conservative extension also is a (proof-theoretic) conservative extension in the above sense.{{Cite book | last1=Hodges | first1=Wilfrid | author1-link=Wilfrid Hodges | title=A shorter model theory | publisher= Cambridge University Press| location=Cambridge | isbn=978-0-521-58713-6 | year=1997 | postscript= | page=58 exercise 8}} 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.

References

External links

[http://www.cs.nyu.edu/pipermail/fom/1998-October/002306.html The importance of conservative extensions for the foundations of mathematics]

Category:Mathematical logic

Category:Model theory

Category:Proof theory

conservative extension

Examples

Model-theoretic conservative extension

See also

References

External links