A NEW COMPUTATION OF THE Σ-ORDINAL OF KPω

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

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Kripke-Platek set theory proof-theoretic ordinals functional interpretations tree functionals strictΠ1- reflection conservativity

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The Journal of Symbolic Logic , Volume 79 , Issue 1 , March 2014 , pp. 306 - 324

DOI: https://doi.org/10.1017/jsl.2013.31

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