On critical cardinalities related to $Q$-sets

Abstract

In this note we collect some known information and prove new results about the small uncountable cardinal q 0 . The cardinal q 0 is defined as the smallest cardinalityWe present a simple proof of a folklore fact that p ≤ q 0 ≤ min{b, non(N ), log(c + )}, and also establish the consistency of a number of strict inequalities between the cardinal q 0 and other standard small uncountable cardinals. This is done by combining some known forcing results. A new result of the paper is the consistency of p < lr < q 0 , where lr denotes the linear refinement number. Another new result is the upper bound q 0 ≤ non(I) holding for any q 0 -flexible cccc σ-ideal I on R.