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- W4254140830 abstract "Let a be an admissible ordinal and let ∧ ≤ α be a limit ordinal. A sequence of a-r.e. degrees is said to be ascending, simultaneous and of length ∧ if (i) there is an α-recursive function t : α × ∧ → α such that, for all ϒ < ∧, A ϒ = { t (σ, ϒ)∣ σ < α} is of degree a ϒ ; (ii) if ϒ < ⊤ < ∧, then a ϒ ≤ α a τ and (iii) for all ϒ < ∧, there is a ⊤ > ϒ with a ϒ , > α a ϒ . Lerman [4] showed that such an exists for every ∧ ≤ α. An upper bound a of is an α-r.e. degree in which every element of is α-recursive. a is minimal if there is no α-r.e. degree b < α a which is also an upper bound of . Sacks [6] proved that every ascending sequence of simultaneously ω-r.e. degrees of length ω cannot have 0 ω ′, the complete ω-r.e. degree, as a minimal upper bound. In contrast, Cooper [2] showed that there exists an ascending sequence of simultaneously ω-r.e. degrees of length to having a minimal upper bound which is an ω-r.e. degree. In this paper we investigate the behavior of ascending sequences of simultaneously α-r.e. degrees for admissible ordinals α > ω. Call α Σ ∞ -admissibIe if it is Σ n -nadmissible for all n . Let Φ(∧) say: No ascending sequence of simultaneously α-r.e. degrees of length ∧ can have 0 α ′, the complete α-r.e. degree, as a minimal upper bound. Our main result in this paper is: Let α be either a constructible cardinal with σ2ci(α) < α or Σ ∞ -admissible. Then σ2cf(α) is the least ordinal ν for which every ∧ ≤ α of cofinality ν (over L α ) can satisfy Φ(∧)." @default.
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- W4254140830 date "1976-03-01" @default.
- W4254140830 modified "2023-09-25" @default.
- W4254140830 title "Minimal upper bounds for ascending sequences of α-recursively enumerable degrees" @default.
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- W4254140830 doi "https://doi.org/10.2307/2272966" @default.
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