Prikry-type forcing and minimal α-degree
release_f4hqccqiejbtxnt46oo5d2l6ie
by
Yang Sen
2013
Abstract
In this paper, we introduce several classes of Prikry-type forcing notions,
two of which are used to produce minimal generic extensions, and the third is
applied in α-recursion theory to produce minimal covers. The first
forcing as a warm up yields a minimal generic extension at a measurable
cardinal (in V), the second at an ω-limit of measurable cardinals
〈γ_n n<ω〉 such that each γ_n (n>0)
carries γ_n-1-many normal measures. Via a notion of V_γ -degree
(see Definition <ref>), we transfer the second Prikry-type
construction for minimal generic extensions to a construction for minimal
degrees in α-recursion theory. More explicitly, Suppose
〈γ_n n<ω〉 is a strictly increasing sequence of
measurable cardinals such that for each n>0, γ_n carries at least
γ_n-1-many normal measures. Let γ={γ_n
n<ω}.
is an A⊂γ such that
(a)
(L_γ,∈,A) is not admissible.
(b) The γ-degree that
contains A has a minimal cover.
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