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In [[mathematical logic]], a [[logical theory]] <math>T_2</math> is a ([[Proof theory|proof theoretic]]) '''conservative extension''' of a theory <math>T_1</math> if the language of <math>T_2</math> extends the language of <math>T_1</math>; every theorem of <math>T_1</math> is a theorem of <math>T_2</math>; and any theorem of <math>T_2</math> that is in the language of <math>T_1</math> is already a theorem of <math>T_1</math>.
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More generally, if Γ is a set of formulas in the common language of <math>T_1</math> and <math>T_2</math>, then <math>T_2</math> is '''Γ-conservative''' over <math>T_1</math> if every formula from Γ provable in <math>T_2</math> is also provable in <math>T_1</math>.
 
To put it informally, the new theory may possibly be more convenient for proving [[theorem]]s, but it proves no new theorems about the language of the old theory.
 
Note that a conservative extension of a [[consistent]] theory is 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, <math>T_0</math>, that is known (or assumed) to be consistent, and successively build conservative extensions <math>T_1</math>, <math>T_2</math>, ... of it.
 
The theorem provers [[Isabelle (theorem prover)|Isabelle]] and [[ACL2]] adopt this methodology by providing a language for conservative extensions by definition.
 
Recently, conservative extensions have been used for defining a notion of [[ontology modularization|module]] for [[Ontology (computer science)|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==
* ACA<sub>0</sub> (a subsystem of [[second-order arithmetic]]) is a conservative extension of first-order [[Peano arithmetic]].
* [[Von Neumann–Bernays–Gödel set theory]] is a conservative extension of [[Zermelo–Fraenkel set theory]] with the [[axiom of choice]] (ZFC).
* [[Internal set theory]] is a conservative extension of [[Zermelo–Fraenkel set theory]] with the [[axiom of choice]] (ZFC).
* [[Extension by definitions|Extensions by definitions]] are conservative.
* Extensions by unconstrained predicate or function symbols are conservative.
* IΣ<sub>1</sub> (a subsystem of Peano arithmetic with induction only for [[arithmetical hierarchy|Σ<sup>0</sup><sub style="margin-left:-0.65em">1</sub>-formulas]]) is a Π<sup>0</sup><sub style="margin-left:-0.65em">2</sub>-conservative extension of the [[primitive recursive arithmetic]] (PRA).
* ZFC is a [[analytical hierarchy|Π<sup>1</sup><sub style="margin-left:-0.65em">3</sub>]]-conservative extension of ZF by [[absoluteness (mathematical logic)|Shoenfield's absoluteness theorem]].
* ZFC with the [[continuum hypothesis]] is a Π<sup>2</sup><sub style="margin-left:-0.65em">1</sub>-conservative extension of ZFC.
 
==Model-theoretic conservative extension==
 
With [[Model theory|model-theoretic]] means, a stronger notion is obtained: an extension <math>T_2</math> of a theory <math>T_1</math> is '''model-theoretically conservative''' if every model of <math>T_1</math> can be expanded to a model of <math>T_2</math>. It is straightforward to see that each model-theoretic conservative extension also is a (proof-theoretic) conservative extension in the above sense. 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.
 
{{Further2|[[Conservativity theorem]]}}
<!--* [[relative consistency]]-->
 
==External links==
*[http://www.cs.nyu.edu/pipermail/fom/1998-October/002306.html The importance of conservative extensions for the foundations of mathematics]
 
[[Category:Proof theory]]
[[Category:Model theory]]

Latest revision as of 02:20, 13 October 2014

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