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RELATIVE PREDICATIVITY AND DEPENDENT RECURSION IN SECOND-ORDER SET THEORY AND HIGHER-ORDER THEORIES

Published online by Cambridge University Press:  18 August 2014

SATO KENTARO
SATO KENTARO*
Affiliation:
INSTITUT FÜR INFORMATIK UND ANGEWANDTE MATHEMATIK, UNIVERSITÄT BERN, CH3012 BERN, SWITZERLAND, E-mail: [email protected]
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Abstract

This article reports that some robustness of the notions of predicativity and of autonomous progression is broken down if as the given infinite total entity we choose some mathematical entities other than the traditional ω. Namely, the equivalence between normal transfinite recursion scheme and new dependent transfinite recursion scheme, which does hold in the context of subsystems of second order number theory, does not hold in the context of subsystems of second order set theory where the universe V of sets is treated as the given totality (nor in the contexts of those of n+3-th order number or set theories, where the class of all n+2-th order objects is treated as the given totality).

Keywords

Relative predicativitydependent transfinite recursionautonomous progression of autonomous progressionsecond order set theoryhigher order number theory
Type
Articles
Information
The Journal of Symbolic Logic , Volume 79 , Issue 3 , September 2014 , pp. 712 - 732
Copyright
Copyright © The Association for Symbolic Logic 2014 

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