Extension by new constant and function names

{{Short description|Mathematical principle}}

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

:$\backslash exists\; x\_1\backslash ldots\backslash exists\; x\_m\backslash ,\backslash varphi(x\_1,\backslash ldots,x\_m)$

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

:$a\_1,\backslash ldots,a\_m$

and adding a new axiom

:$\backslash varphi(a\_1,\backslash ldots,a\_m)$.

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 $T$.

Such a theory can also be conservatively extended by introducing a new functional symbol:{{cite book |last1=Shoenfield |first1=Joseph |authorlink = Joseph Shoenfield|title=Mathematical Logic |date=1967 |pages=55–56

|publisher=Addison-Wesley}}

Suppose that a closed formula $\backslash forall\; \backslash vec\{x\}\backslash ,\backslash exists\; y\backslash ,\backslash !\backslash ,\backslash varphi(y,\backslash vec\{x\})$ is a theorem of a first-order theory $T$, where we denote $\backslash vec\{x\}:=(x\_1,\backslash ldots,x\_n)$. Let $T\_1$ be a theory obtained from $T$ by extending its language with a new functional symbol $f$ (of arity $n$) and adding a new axiom $\backslash forall\; \backslash vec\{x\}\backslash ,\backslash varphi(f(\backslash vec\{x\}),\backslash vec\{x\})$. 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 $f$).

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.