Conservativity theorem

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

is a theorem of a first-order theory . Let be a theory obtained from by extending its language with new constants

and adding a new axiom

.

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

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

References[edit]

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