Content deleted Content added
→Other similar OCFs > Arai's ψ: new section |
|||
Line 78:
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)
| |||