Inmathematical logic, atheory can be extended withnew constants or function names under certain conditions with assurance that the extension will introduceno contradiction.Extension by definitions is perhaps the best-known approach, but it requiresunique existence of an object with the desired property. Addition of new names can also be donesafely without uniqueness.

Suppose that aclosed formula

${\displaystyle \mathrm{\exists }{x}_1\dots \mathrm{\exists }{x}_m\phi (x_1,\dots ,x_m)}$

is a theorem of afirst-order theory${\displaystyle T}$. Let${\displaystyle T_1}$ be a theory obtained from${\displaystyle T}$ by extending itslanguage with new constants

${\displaystyle a_1,\dots ,a_m}$

and adding a newaxiom

${\displaystyle \phi (a_1,\dots ,a_m)}$.

Then${\displaystyle T_1}$ is aconservative extension of${\displaystyle T}$, which means that the theory${\displaystyle T_1}$ has the same set of theorems in the original language (i.e., without constants${\displaystyle a_i}$) as the theory${\displaystyle T}$.

Such a theory can also be conservatively extended by introducing a newfunctional symbol:^[1]

Suppose that aclosed formula${\displaystyle \mathrm{\forall }\overrightarrow{x}\mathrm{\exists }y\phi (y,\overrightarrow{x})}$ is a theorem of a first-order theory${\displaystyle T}$, where we denote${\displaystyle \overrightarrow{x}:=(x_1,\dots ,x_n)}$. Let${\displaystyle T_1}$ be a theory obtained from${\displaystyle T}$ by extending its language with a new functional symbol${\displaystyle f}$ (of arity${\displaystyle n}$) and adding a new axiom${\displaystyle \mathrm{\forall }\overrightarrow{x}\phi (f(\overrightarrow{x}),\overrightarrow{x})}$. Then${\displaystyle T_1}$ is aconservative extension of${\displaystyle T}$, i.e. the theories${\displaystyle T}$ and${\displaystyle T_1}$ prove the same theorems not involving the functional symbol${\displaystyle f}$).

Shoenfield states the theorem in the form for a new function name, and constants are the same as functionsof 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.

• Conservative extension
• Extension by definition

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