9 research outputs found
A model in which the Separation principle holds for a given effective projective Sigma-class
In this paper, we prove the following: If , there is a generic
extension of -- the constructible universe -- in which it is true that the
Separation principle holds for both effective (lightface) classes
and for sets of integers. The result was announced
long ago by Leo Harrington with a sketch of the proof for ; its full proof
has never been presented. Our methods are based on a countable product of
almost-disjoint forcing notions independent in the sense of Jensen--Solovay.Comment: 17 page
Non-uniformizable sets with countable cross-sections on a given level of the projective hierarchy
We present a model of set theory, in which, for a given , there exists
a non-ROD-uniformizable planar lightface set in , whose all vertical cross-sections are countable sets (and in
fact Vitali classes), while all planar boldface sets with
countable cross-sections are -uniformizable. Thus it is true
in this model, that the ROD-uniformization principle for sets with countable
cross-sections first fails precisely at a given projective level.Comment: A revised version of the originally submitted preprin