Abstract
Given a binary relation R, we call a subset A of the range of R R-adequate if for every x in the domain there is some y ∈ A such that (x, y) ∈ R. Following Blass, we call a real n "needed" for R if in every A-adequate set we find an element from which n is Turing computable. We show that every real needed for inclusion on the Lebesgue null sets, Cof(N), is hyperarithmetic. Replacing "R-adequate" by "R-adequate with minimal cardinality" we get the related notion of being "weakly needed". We show that it is consistent that the two notions do not coincide for the reaping relation. (They coincide in many models.) We show that not all hyperarithmetic reals are needed for the reaping relation. This answers some questions asked by Blass at the Oberwolfach conference in December 1999 and in [4].
Original language | English |
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Pages (from-to) | 1-37 |
Number of pages | 37 |
Journal | Israel Journal of Mathematics |
Volume | 141 |
DOIs | |
State | Published - 2004 |