Minimal upper bounds for sequences of -degrees
Alexander S. Kechris · Journal of Symbolic Logic · 1978
It is proved here, assuming Projective Determinacy, that every ascending sequence of -degrees has a minimal strict upper bound but no least strict upper bound. This generalizes a result of Friedman for n = 1. Our general notation and terminology will be that of [Ke1] and [Mo1]. Letters i,j, k,… denote members of ω and α, β, ϒ,… members of ωω i.e. reals. Projective Determinacy (PD) is the hypothesis that every projective set of reals in determined, while in general for a collection of sets of reals Γ, Determinacy (Γ) abbreviates the statement that every set in Γ is determined. §1. -degrees. For each m ≥ 1 and α, β Є ωω let α ≤mβ ⇔ α Є (β), α