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

:<math>\exists x_1\ldots\exists x_m\,\varphi(x_1,\ldots,x_m)</math>

is a theorem of a [[first-order theory]] <math>T</math>. Let <math>T_1</math> be a theory obtained from <math>T</math> by extending its [[formal language|language]] with new constants

:<math>a_1,\ldots,a_m</math>

and adding a new [[axiom]]

:<math>\varphi(a_1,\ldots,a_m)</math>.

Then <math>T_1</math> is a [[conservative extension]] of <math>T</math>, which means that the theory <math>T_1</math> has the same set of theorems in the original language (i.e., without constants <math>a_i\,\!</math>) as the theory <math>T</math>.

In a more general setting, the '''conservativity theorem''' is formulated for extensions of a first-order theory by introducing a new [[Functional predicate|functional symbol]]:

:Suppose that a ''closed'' formula <math>\forall \vec{y}\,\exists x\,\!\,\varphi(x,\vec{y})</math> is a theorem of a first-order theory <math>T</math>, where we denote <math>\vec{y}:=(y_1,\ldots,y_n)</math>. Let <math>T_1</math> be a theory obtained from <math>T</math> by extending its language with new functional symbol <math>f\,\!</math> (of arity <math>n</math>) and adding a new axiom <math>\forall \vec{y}\,\varphi(f(\vec{y}),\vec{y})</math>. Then <math>T_1</math> is a [[conservative extension]] of <math>T</math>, i.e. the theories <math>T</math> and <math>T_1</math> prove the same theorems not involving the functional symbol <math>f\,\!</math>).

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

{{logic-stub}}
[[Category:Mathematical logic]]
[[Category:Theorems in the foundations of mathematics]]
[[Category:Proof theory]]
{{mathlogic-stub}}

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