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Informally, the new theory may possibly be more convenient for proving Theorem s, but it proves no new theorems about the old theory. The importance of this notion lies in the following theorem: : If T2 is a conservative extension of T1, and T1 is consistent, then T2 is consistent as well. 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. The theorem provers 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 Module for 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. Examples
MODEL-THEORETIC CONSERVATIVE EXTENSION With Model-theoretic means, a stronger notion is obtained: T2 is a model-theoretic conservative extension of T1 if every model of T1 can be expanded to a model of T2. 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. ''See also'': Conservativity Theorem EXTERNAL LINKS |
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