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A NEW COMPUTATION OF THE Σ-ORDINAL OFKPω

Published online by Cambridge University Press:  17 April 2014

FERNANDO FERREIRA
FERNANDO FERREIRA*
Affiliation:
DEPARTAMENTO DE MATEMÁTICA FACULDADE DE CIÊNCIAS UNIVERSIDADE DE LISBOA CAMPO GRANDE, ED. C6, 1749-016 LISBOA, PORTUGAL E-mail: fjferreira@fc.ul.pt
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Abstract

We define a functional interpretation of KPω using Howard’s primitive recursive tree functionals of finitetype and associated terms. We prove that the Σ-ordinal of KPω is the least ordinal not given by a closed term of the ground type of thetrees (the Bachmann-Howard ordinal). We also extend KPω to a second-order theory withΔ 1-comprehension and strict- ${\rm{\Pi }}_1^1$ reflection and show that the Σ-ordinal of thistheory is still the Bachmann-Howard ordinal. It is also argued that thesecond-order theory is Σ1-conservative overKPω.

Keywords

Kripke-Platek set theoryproof-theoretic ordinalsfunctional interpretationstree functionalsstrictΠ1- reflectionconservativity

Information

Type
Articles
Information
The Journal of Symbolic Logic , Volume 79 , Issue 1 , March 2014 , pp. 306 - 324
Copyright
Copyright © Association for Symbolic Logic 2014 

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