 The third clause in this definition defines the value of \(\pi(i,x)\) in terms of the value \(\pi(i,x-1)\) rather than \(\pi(i-1,x-1)\) in analogy with the scheme (\ref{gprimrec}). It may thus not be immediately obvious that the definition (\ref{pidef}) describes an algorithm for computing the values of \(\pi(i,x)\) which always terminates in the manner illustrated by the calculation (\ref{factcalc}). Note, however, with each recursive application either \(i\) decreases, or \(i\) remains the same and \(x\) decreases. It thus follows that each time \(x\) reaches 0, \(i\) will start to decrease so that the base case is eventually reached. Thus although the value of \(\pi(i,x)\) grows very rapidly—e.g., \(\pi(4,3) = 2^{2^{65536}}-3\)—it is still reasonable to regard (\ref{pidef}) as satisfying Gödel’s requirement that a recursively defined function is computable by a finite procedure.