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An ordinal analysis of parameter free Π12-comprehension

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Abstract.

This paper is the second in a series of three culminating in an ordinal analysis of Π12-comprehension. Its objective is to present an ordinal analysis for the subsystem of second order arithmetic with Δ12-comprehension, bar induction and Π12-comprehension for formulae without set parameters. Couched in terms of Kripke-Platek set theory, KP, the latter system corresponds to KPi augmented by the assertion that there exists a stable ordinal, where KPi is KP with an additional axiom stating that every set is contained in an admissible set.

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Authors and Affiliations

  1. Department of Pure Mathematics, University of Leeds, Leeds, LS2 9JT, UK

    Michael Rathjen

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  1. Michael Rathjen
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Correspondence to Michael Rathjen.

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Es wurden nun des längeren die einfachsten Modellherleitungen gewertet, mit der Ausweitungsidee. Es ergaben sich transfinite Stufen, verschieden je nach Geschick. Unerfreuliche technische Mühsal. Die Stufen gingen sehr rasch in die Höhe. [...] So etwas ist durchaus denkbar: Steigerung der Vielfältigkeit der erforderlichen mitzuschleppenden Funktionen mit dem Sequenzgrade. Es hängt sachlich alles vielfältig miteinander zusammen, und der Konstruktivist hat die Aufgabe, alle diese Zusammenhänge auseinander zu klauben und zum Ausdruck zu bringen. [...] Man muß den mühsamen Weg der Einzeluntersuchung gehen, des schrittweisen Aufstieges, einen Begriff nach dem anderen hinzunehmend. GERHARD GENTZEN (1945) ([9])

In the main this research was carried out in 1995. The particular presentation, however, is a more recent development.

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Rathjen, M. An ordinal analysis of parameter free Π12-comprehension. Arch. Math. Logic 44, 263–362 (2005). https://doi.org/10.1007/s00153-004-0232-4

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