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== wall street ==
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Please see [[User:JRSpriggs/Ordinal notation]]. [[User:JRSpriggs|JRSpriggs]] ([[User talk:JRSpriggs|talk]]) 18:41, 2 April 2022 (UTC)
== 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)
:Also, to add on to my previous comment:
:In the section ‘collapsing large cardinal’, the following is stated about the PTO of KPi:
:"Roughly speaking, this collapse can be obtained by adding the α ↦ Ω_α function itself to the list of constructions to which the C(·) collapsing system applies."
:This is wrong in two different ways. Firstly, adding α ↦ Ω_α to the construction does not increase the strength of the ordinal notation that much as you'd have gaps between ε{Ω_α+1} and Ω_{α+1} for each positive α, thus one should also make a collapsing function for each Ω_α as done in ‘Going beyond the BHO’. Secondly, and more importantly, this approach would only reach the extended Buchholz ordinal (collapse of Φ₁(0)) when done right, which is way below the proof theoretic ordinal of KPi. To get to the proof theoretic ordinal of KPi one must use a large cardinal I (the least weakly inaccessible) used in a collapsing function diagonalizing over Ω_α in the same way that the main OCF on the article diagonalizes over ε_α.
:It is also stated that the PTO of KPi is the collapse of the least weakly inaccessible cardinal. In the article, this is stated confusingly and I think it should be clarified what is meant exactly. The collapse of a weakly inaccessible cardinal is not the PTO of KPi, the collapse of the least ε-number after the least weakly inaccessible is. Same for KPM and Π₃-ref.
:I'll probably make edits to the article after a few days if no-one comments to disagree that changes to the article with respect to what is listed above should be made. [[Special:Contributions/213.93.13.10|213.93.13.10]] ([[User talk:213.93.13.10|talk]]) 20:05, 3 March 2024 (UTC)
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