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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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The Journal of Symbolic Logic
  • ISSN: 0022-4812
  • EISSN: 1943-5886
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