Mathematical logic | Model theory | Proof theory
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 is a (proof theoretic) conservative extension of a theory if every theorem of is a theorem of , and any theorem of in the language of is already a theorem of . More generally, if is a set of formulas in the common language of and , then is -conservative over if every formula from provable in is also provable in . 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 would be a theorem of , so every formula in the language of would be a theorem of , so 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, , that is known (or assumed) to be consistent, and successively build conservative extensions , , ... of it. 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. An extension which is not conservative may be called a proper extension. (Wikipedia).
Is There an Alternative to Political Correctness?
Political correctness aims for some very nice results, but its means have a habit of upsetting a lot of people. Might there be an alternative to it? We think there is, and it’s called Politeness. If you like our films, take a look at our shop (we ship worldwide): https://goo.gl/iVqWJ1 Joi
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