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:\exists x_1\ldots\exists x_m\, arphi(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

: arphi(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 orall ec{y}\,\exists x\,\!\, arphi(x, ec{y}) is a theorem of a first-order theory T, where we denote ec{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 orall ec{y}\, arphi(f( ec{y}), ec{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\,\!).


BIBLIOGRAPHY

  • Elliott Mendelson ( 1997 ). ''Introduction to Mathematical Logic'' (4th ed.) Chapman & Hall.

  • J.R. Shoenfield ( 1967 ). ''Mathematical Logic''. Addison-Wesley Publishing Company.