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== Possible error in one of the example ψ-functions?==
Before the simultaneous recursion is introduced, there is a circular reference via ψ<sub>1</sub>. ψ<sub>0</sub>(α) is defined using a set C<sub>0</sub>(α), which includes the ordinal ψ<sub>0</sub>(α+2). However, ψ<sub>1</sub>(α+2) is defined using a set C<sub>1</sub>(α+2), which contains ψ<sub>0</sub>(α0+1), which means ψ is defined by a non-terminating inductive clause. So I think this version of ψ is not just hard to extend as mentioned in the article, but not a well-defined system of functions. I'm not sure how it should be fixed while keeping its current purpose, an intermediate strengthening of ψ which is meant to lead into the system with arbitrarily many ψ-functions. [[User:C7XWiki|C7XWiki]] ([[User talk:C7XWiki|talk]]) 22:13, 25 September 2022 (UTC)
== Other similar OCFs > Arai's ψ ==
In the article, it is said that ψΩ(ε{Ι+1}) is the TFBO in the section of Arai's ψ function, where I denotes the least recursively inaccessible. This is wrong. The TFBO is the proof theoretic ordinal of ID_ω and the recursive collapse of ε{Ω_ω+1} with respect to an appropriate ordinal collapsing function, Ω_ω representing either the least limit of infinite cardinals or least limit of admissibles depending on if recursive or non-recursive analogues are used. ψΩ(ε{I+1}) is the proof theoretic ordinal of KPi (sometimes denoted with capital I) and is much larger than the TFBO.
The confusion between the TFBO and the proof theoretic ordinal of KPi probably comes from the fact that KPi = KPI and KPl have similar names, while one (KPi) has a proof theoretic ordinal of ψΩ(ε{I+1}) while the other (KPl) has the TFBO as proof theoretic ordinal. One has probably confounded these two theories and has mistakenly said that ψΩ(ε{I+1}) is the TFBO.
Should this be changed? [[Special:Contributions/213.93.13.10|213.93.13.10]] ([[User talk:213.93.13.10|talk]]) 13:54, 3 March 2024 (UTC)
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