User blog comment:Ynought/My attempt at an ordinal function/@comment-35470197-20190313221317/@comment-35470197-20190315233658

It does not work because it is unable to reconstruct \(\eta\) and \(\alpha\) from the ordinal \(\eta \alpha\). For example, consider the following two functions: \begin{eqnarray*} \eta_0 \alpha & = & \alpha \\ \eta_1 \alpha & = & \aleph_{\alpha} \end{eqnarray*} Then your rules imply the following two distinct ways to compute \(\Theta(\Omega)\) : \begin{eqnarray*} \Theta(\Omega) & = & \Theta(\eta_0 \Omega) = \sup(\Theta(\Theta(\eta_0 \omega)),\ldots) \\ \Theta(\Omega) & = & \Theta(\eta_1 1) = \sup(\Theta(\eta_1 1[0]),\Theta(\eta_1 1[1]),\ldots) \end{eqnarray*} Another serious point is that \(\eta \alpha[n]\) is not defined for a general case, because there is not a unique way to express \(\eta \alpha\) as \(\Theta(\beta)\) for an ordinal \(\beta\). For example, \(\eta_1 1[n]\) above is not well-defined.

Generally, using functions (not function symbols) in this way does not yield valid ordinal functions.