Collapsing functions based on recursively large ordinals: A well-ordering proof for KPM (Q1322453)

This paper continues the proof-theoretical investigation of KPM started in an earlier paper of the author [ibid. 30, No. 5/6, 377-403 (1991; Zbl 0727.03036)], stating that \(|\text{KPM}|\leq\psi(\chi 00)(\psi(\chi\varepsilon_{M+1}0)0)\). In order to determine a lower bound of \(|\text{KPM}|\), a previously developed ordinal denotation system TM [the author, ibid. 29, No. 4, 249-263 (1990; Zbl 0709.03042)] is transferred into KPM. This shows how to replace large cardinals by their recursive counterparts in the development of denotation systems. The most delicate part of this project is to establish the collapsing property of certain functions which could be done in the original system just by cardinality arguments now not available. In a very detailed exposition the author shows how to use in KPM the properties of admissible ordinals and recursive Mahlo cardinals to overcome this restriction. By embedding proper subsystems TMM of TM into \(\widetilde C_{\widetilde Z(0)}(\widetilde Z(\widetilde M(m)))\) (with respect to KPM) he works out that for any initial segment of \(\psi(\chi 00)(\psi(\chi\varepsilon_{M+1}0)0)\) the well-ordering of \(\langle\text{TMM},\leq\rangle\) can be derived in KPM. Therefore it follows that \(|\text{KPM}|\geq \psi(\chi 00)(\psi(\chi\varepsilon_{M+1}0)0)\), and the proof-theoretical ordinal of KPM is completely determined.