Conservativity theorem

In mathematical logic, the conservativity theorem states the following: Suppose that a closed formula

\exists x_1\ldots\exists x_m\,\varphi(x_1,\ldots,x_m)

is a theorem of a first-order theory $T$ . Let $T_{1}$ be a theory obtained from $T$ by extending its language with new constants

a_{1},\ldots ,a_{m}

and adding a new axiom

\varphi(a_1,\ldots,a_m)

.

Then $T_{1}$ is a conservative extension of $T$ , which means that the theory $T_{1}$ has the same set of theorems in the original language (i.e., without constants $a_{i}$ ) 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

\forall \vec{y}\,\exists x\,\!\,\varphi(x,\vec{y})

is a theorem of a first-order theory

T

, where we denote

\vec{y}:=(y_1,\ldots,y_n)

. Let

T_{1}

be a theory obtained from

T

by extending its language with a new functional symbol

f

(of arity

n

) and adding a new axiom

\forall \vec{y}\,\varphi(f(\vec{y}),\vec{y})

. Then

T_{1}

is a conservative extension of

T

, i.e. the theories

T

and

T_{1}

prove the same theorems not involving the functional symbol

f

).

References

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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