 it is easy to see that this  function also cannot be primitive recursive. For if \(\delta(x)\) coincided with some function \(g_j(x)\) in the enumeration, then we would have \(g_j(j) = \delta(j) = g_j(j) + 1\), a contradiction. Note that this also shows that relative to such an enumeration the universal function \(u_1(i,x) = g_i(\vec{x})\) for unary primitive recursive functions cannot itself be primitive recursive as we could otherwise define \(\delta(x)\) as \(u_1(x,x) + 1\). Hilbert &amp; Bernays (1939, ch. 5) would later discuss this observation in regard to what has become known as their denotational paradox—see, e.g., (Priest 1997).