User blog comment:Denis Maksudov/Slowly growing ordinal function and FS up to BHO./@comment-5529393-20170402194355/@comment-28606698-20170402222719

That is exactly what I meant. As you, may be, remembered, in previous version of section for FS I used some non-elegant grammatical construction: last term in NF for argument or last term in NF for exponent of last term in NF for argument if the exponent is uncountable... and so on - I will not write it fully. Any way, that is not good for definition.

Probably we can use next definition:

$$C_0(\alpha)=\{0,\Omega\}$$

$$C_{n+1}(\alpha)=\{\psi'(\eta),\beta+\gamma,\xi^{\gamma},\xi \gamma|\alpha,\beta,\gamma,\eta,\xi\in C_n(\alpha),\eta<\alpha,\xi\geq\Omega\}$$

$$C(\alpha)=\cup_{n=0}^\omega C_n(\alpha)$$

$$\psi'(\alpha)=min\{\beta<\Omega|\beta\notin C(\alpha)\}$$

One difference with traditional construction of collapsing function: only uncountable ordinal can be exponentiated, becouse I want to obtain function, which allows to express any non-zero ordinal and that's why we need $$\psi'(0)=1$$, not $$\varepsilon$$.