Prior to this Bennett (1962) had shown that there exists a \(\Delta_0\)-formula \(\varepsilon(x,y,z)\) which defines the graph of the exponentiation function relative to the standard model of arithmetic. \(\text{I}\Delta_0\) proves that \(\varepsilon(x,y,z)\) satisfies the standard defining equations for exponentiation. But it may also be shown that this theory does not prove the totality of exponentiation under this or any other definition in the following sense: