In mathematical logic, a theory can be extended with new constants or function names under certain conditions with assurance that the extension will introduce no contradiction. Extension by definitions is perhaps the best-known approach, but it requires unique existence of an object with the desired property. Addition of new names can also be done safely without uniqueness.
Suppose that a closed formula
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 .
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 a 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 ).
Shoenfield states the theorem in the form for a new function name, and constants are the same as functions of zero arguments. In formal systems that admit ordered tuples, extension by multiple constants as shown here can be accomplished by addition of a new constant tuple and the new constant names having the values of elements of the tuple.
- Shoenfield, Joseph (1967). Mathematical Logic. Addison-Wesley. pp. 55–56.