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(1999)
Bar induction and ω model reflection
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Translated from the revised German edition by
SG Simpson (2009)
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Proof Theory: The First Step into Impredicativity
If ξ = ϑ(ξ ), then K (ξ ) = {ξ }, hence this also trivially holds. Assume ξ = Ω n ξ n + · · · + Ω 0 ξ 0 with Ω > ξ n > 0 and Ω > ξ 1 , . . . , ξ −1
Lemma 10 If ξ ∈ OT (ϑ), then K (ξ ) ⊆ OT (ϑ). Furthermore, G ϑ (k(ξ )) ≤ G ϑ (ξ ) for all ξ in O T (ϑ)
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In this article we provide an intrinsic characterization of the famous Howard–Bachmann ordinal in terms of a natural well-partial-ordering by showing that this ordinal can be realized as a maximal order type of a class of generalized trees with respect to a homeomorphic embeddability relation. We use our calculations to draw some conclusions about some corresponding subsystems of second order arithmetic. All these subsystems deal with versions of light-face $$\varPi ^1_1$$ Π 1 1 -comprehension.
Archive for Mathematical Logic – Springer Journals
Published: Nov 5, 2016
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