In mathematical logic, the conservativity theorem states the following: Suppose that a closed formula
and adding a new axiom
Then is a conservative extension of , which means that the theory has the same set of theorems in the original language (i.e., without constants ) as the theory .
In a more general setting, the conservativity theorem is formulated for extensions of a first-order theory by introducing a new functional symbol:
- Suppose that a closed formula is a theorem of a first-order theory , where we denote . Let be a theory obtained from by extending its language with a new functional symbol (of arity ) and adding a new axiom . Then is a conservative extension of , i.e. the theories and prove the same theorems not involving the functional symbol ).
- Elliott Mendelson (1997). Introduction to Mathematical Logic (4th ed.) Chapman & Hall.
- J.R. Shoenfield (1967). Mathematical Logic. Addison-Wesley Publishing Company.
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