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I am reading about fast growing hierarchies and the ordinal functions that are used in the indices, which allow us to categorise fast growing functions by selecting fundamental sequences for these large countable ordinals. For instance the Veblen functions, or the ordinal collapsing function $\psi$. The countability of all of these ordinals is obvious if we use $\mathsf{AC}_{\omega}$ ($\Rightarrow$ a countable of union of countable sets is countable). My question is, how does one prove the same statements in $\mathsf{ZF}?$ (for instance, for large countable ordinals arising from these functions)

Asaf Karagila
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Sonk
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  • I have no idea about how to provide an exact answer, but I bet your ordinals (that are given by Veblen functions or ordinal collapsing functions) are all recursive, so countable. – Hanul Jeon Oct 13 '20 at 17:28
  • https://math.stackexchange.com/questions/1285395/how-to-show-countability-of-omega-omega-or-epsilon-0-in-zf and https://math.stackexchange.com/questions/181424/varepsilon-number-countability-without-choice might be helpful to answer your question. – Asaf Karagila Oct 13 '20 at 20:19

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The countability result is often proven differently depending on the definition, although unfortunately I'm not sure the proofs given in the original published literature are choiceless. When the definition of the ordinal collapsing function is simply based on the minimum excludant from one of the closure sets (like in Buchholz's "A new system of proof-theoretic ordinal functions", where $\psi_\nu(\alpha)=\textrm{min}\{\beta\mid\beta\notin C_\nu(\alpha)\}$) often a cardinality argument is used - for example Buchholz uses the fact that $C_0(\alpha)$'s cardinality is less than $\Omega$ to show that there must exist $\beta<\Omega$ where $\beta\notin C_0(\alpha)$, so $\psi_0(\alpha)<\Omega$.

For definitions not just based on the minimum excludant, at least for definitions appearing in literature there seems to be another common general form of them, similar to this sketch on Buchholz's "A simplified version of local predicativity" (full definition of $\psi_\pi(\alpha)$ for $\pi\in\textrm{Reg}\cap I$ on p.140):

We have a binary $C(\alpha,\beta)\supseteq\beta$, and $\psi_\Omega(\alpha)$ is the least $\beta$ where $C(\alpha,\beta)\cap\Omega\subseteq\beta$. Buchholz proves each $\psi_\Omega(\alpha)$ is countable (lemma 4.5.a), by showing that setting

\begin{align*} \beta_0 & = 0 \\ \beta_{n+1} & = \min\{\eta\mid C(\alpha,\beta_n)\cap\Omega\subseteq \eta\} \\ \beta & = \textrm{sup}\{\beta_n\mid n<\omega\} \end{align*}

gives us the necessary $C(\alpha,\beta)\cap\Omega\subseteq\beta$ condition, so $\psi_\Omega(\alpha)\leq\beta$. Unfortunately I don't think this is choiceless, since $\Omega$ may not be regular without choice. However, it may be possible to prove countability by passing to a version of the collapsing function which uses recursive analogues and has the same non-minimum-excludant form (see Rathjen's "How to develop proof-theoretic ordinal functions on the basis of admissible ordinals", Math. Logic Quarterly vol. 39), then show that the $\psi_{\omega_1^{CK}}(\alpha) < \omega_1^{CK}$ property of this function translates back into $\psi_\Omega(\alpha) < \Omega$ in the original case.

C7X
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