 in the manner which extends the notation scheme for partial recursive function introduced at the end of  Section 2.1.1.  By associating the atomic expressions of this language with natural numbers in the manner of Gödel numbering \(\ulcorner \cdot \urcorner\) described in  Section 1.3  and then employing the coding machinery described at the end of  Section 2.1.2,  it is then possible to associate \(\tau\) with a natural number \(\ulcorner \tau \urcorner = e\) which can serve as an index for \(f(\vec{x})\).