and every model of can be expanded to a model of . 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 Extension by definitions Extension by new constant and function names References External links The importance of conservative extensions for the foundations of mathematics Mathematical logic Model theory Proof theory