### On needed reals

by Mildenberger and Shelah. [MdSh:725]

Israel J Math, 2004

Following Blass, we call a real a ``needed'' for a binary
relation R on the reals if in every R-adequate set we find an
element from which a is Turing computable. We show that every real
needed for Cof (N) is hyperarithmetic. Replacing
``R-adequate'' by ``R-adequate with minimal cardinality'' we get
related notion of being ``weakly needed''. We show that is is
consistent that the two notions do not coincide for the reaping
relation. (They coincide in many models.) We show that not all
hyperarithmetical reals are needed for the reaping relation. This
answers some questions asked by Blass at the Oberwolfach conference
in December 1999.

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