User blog comment:Rgetar/Veblen-like collapsing function up to Buchholz ordinal/@comment-35470197-20181128093835

> \(\varphi^{\alpha}_{\beta+1+\gamma}(X) = \varphi^{\alpha}_{\beta}(\varphi^{\alpha+1+\beta}_{\gamma}(X))\)

A presentation of a given ordinal by an expression like \(\beta+1+\gamma\) is not unique, and hence this recursion does not give a well-defined system of functions.

> \(\varphi^{\alpha}(X) = \varphi^{\alpha}_0(X)\)

Are you defining the left hand side as the right hand side? Then the right hand side is not well-defined. Otherwise, the left hand side is not well-defined.

> \(\varphi_{\alpha}(X) = \varphi^0_{\alpha}(X)\)

The same as the comment above.