Conservative extension

• \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}. • 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). • \mathsf{ZFC} is a \Pi^1_4-conservative extension of \mathsf{ZF} by Shoenfield's absoluteness theorem. • \mathsf{ZFC} with the generalized continuum hypothesis is a \Pi^2_1-conservative extension of \mathsf{ZFC}. ==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. 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==