Prikry-type forcing and minimal α-degree release_f4hqccqiejbtxnt46oo5d2l6ie

by Yang Sen

Released as a article .

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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Date   2013-10-03
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arXiv  1310.0891v1
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