Content deleted Content added
Removed ψ_Ω(Ω) from the list of example values at Arai's ψ as it was wrongfully stated to be ε_0. The Skolum hull used in the definition of Arai's ψ is closed under the binary Veblen function, ψ_Ω(Ω) would thus be equal to φ(1,1,0) = β_0. |
→Collapsing large cardinals: we do not need this time-specific wording |
||
Line 320:
* Rathjen<ref>Rathjen, 1994 (Ann. Pure Appl. Logic)</ref> later described the collapse of a [[weakly compact cardinal]] to describe the ordinal-theoretic strength of Kripke–Platek set theory augmented by certain [[reflection principle]]s (concentrating on the case of <math>\Pi_3</math>-reflection). Very roughly speaking, this proceeds by introducing the first cardinal <math>\Xi(\alpha)</math> which is <math>\alpha</math>-hyper-Mahlo and adding the <math>\alpha \mapsto \Xi(\alpha)</math> function itself to the collapsing system.
* In a 2015 paper, Toshyasu Arai has created ordinal collapsing functions <math>\psi^{\vec \xi}_\pi</math> for a vector of ordinals <math>\xi</math>, which collapse <math>\Pi_n^1</math>-[[Indescribable cardinal|indescribable cardinals]] for <math>n>0</math>. These are used to carry out [[ordinal analysis]] of Kripke–Platek set theory augmented by <math>\Pi_{n+2}</math>-reflection principles. <ref>T. Arai, [https://arxiv.org/abs/1907.07611v1 A simplified analysis of first-order reflection] (2015).</ref>
* Rathjen has
== Notes ==
| |||