Well-ordering Principles, ω-models and $$ \varPi_{1}^{1} $$ -comprehension

Michael Rathjen () and Ian Alexander Thomson ()
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Michael Rathjen: University of Leeds, School of Mathematics
Ian Alexander Thomson: University of Leeds, School of Mathematics

Chapter Chapter 12 in The Legacy of Kurt Schütte, 2020, pp 171-215 from Springer

Abstract: Abstract In this paper we show that over the weak theory RCA0 the existence of ω-models of $$ \varPi_{1}^{1} $$ -comprehension is equivalent to saying that a certain elementary endomorphism of the category of linear orders preserves well-orderings. This result continues a long line of research. Technically, among other methods, we use Schutte’s search trees and a relativized version of the ordinal analysis of $$ \varPi_{1}^{1} $$ -comprehension via the Ωn-rules akin to the presentation in Buchholz and Schutte’s 1988 book [3]. A crucial challenge one faces is to show that the latter approach can be made to work in a weak background theory.

Keywords: Reverse mathematics; well ordering principles; Schutte deduction chains; countable coded ω-model; $$ \varPi_{1}^{1} $$ -comprehension; Ωn-rule; well-ordering proof; 03B30; 03F05; 03F15; 03F35; 03F35 (search for similar items in EconPapers)
Date: 2020
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Persistent link: https://EconPapers.repec.org/RePEc:spr:sprchp:978-3-030-49424-7_12

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DOI: 10.1007/978-3-030-49424-7_12

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