In mathematical logic, the conservativity theorem states the following: Suppose that a closed formula Mathematical logic is a discipline within mathematics, studying formal systems in relation to the way they encode intuitive concepts of proof and computation as part of the foundations of mathematics. ...

is a theorem of a first-order theory T. Let T₁ be a theory obtained from T by extending its language with new constants First-order predicate calculus or first-order logic (FOL) is a theory in symbolic logic that permits the formulation of quantified statements such as there is at least one X such that. ...

and adding a new axiom In epistemology, an axiom is a self-evident truth upon which other knowledge must rest, from which other knowledge is built up. ...

.

Then T₁ is a conservative extension of T, which means that the theory T₁ has the same set of theorems in the original language (i.e., without constants ) as the theory T.

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 T, where we denote . Let T₁ be a theory obtained from T by extending its language with new functional symbol (of arity n) and adding a new axiom . Then T₁ is a conservative extension of T, i.e. the theories T and T₁ prove the same theorems not involving the functional symbol ).

References

• Elliott Mendelson (1997). Introduction to Mathematical Logic (4th ed.) Chapman & Hall.
• J.R. Shoenfield (1967). Mathematical Logic. Addison-Wesley Publishing Company.