Equivalence of families of functions on the natural numbers

Abstract:

We present some consequences of the inequality $\mathfrak {u} < \mathfrak {g}$ among cardinal invariants of the continuum, which has previously been shown to be consistent relative to ZFC. We are interested in its effect on two orderings of families of functions on the natural numbers; in particular we show that, under $\mathfrak {u} < \mathfrak {g}$, there are exactly five equivalence classes for both orderings (excluding the families bounded by a fixed constant function). This implies, under the same hypothesis, the existence of exactly four classes of rarefaction of measure zero sets.