Zeroth Hyperoperation is Successor Function

Theorem

The zeroth hyperoperation is the successor function:

$H_0 \left({x, y}\right) = y + 1$

Proof

Immediate by definition of the successor function $s: \Z_{\ge 0} \to \Z_{\ge 0}$:

$\forall y \in \Z_{\ge 0}: s \left({y}\right) = y + 1$

and for the $n$th hyperoperation:

$\forall n, x, y \in \Z_{\ge 0}: H_n \left({x, y}\right) = \begin{cases} y + 1 & : n = 0 \\ x & : n = 1, y = 0 \\ 0 & : n = 2, y = 0 \\ 1 & : n > 2, y = 0 \\ H_{n - 1} \left({x, H_n \left({x, y - 1}\right)}\right) & : n > 0, y > 0 \end{cases}$

setting $n = 0$.

Thus the zeroth hyperoperation degenerates to a mapping with a single operand.

$\blacksquare$