Conservativity theorem

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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 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 $f\,\!$ ).

[edit] Bibliography

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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