Cardinal invariants and the collapse of the continuum by Sacks forcing

Abstract

We study cardinal invariants of systems of meager hereditary families of subsets of ω connected with the collapse of the continuum by Sacks forcing 𝕊 and we obtain a cardinal invariant 𝔥_ω such that 𝕊 collapses the continuum to 𝔥_ω and 𝔥≤𝔥_ω≤𝔟. Applying the Baumgartner-Dordal theorem on preservation of eventually narrow sequences we obtain the consistency of 𝔥=𝔥_ω<𝔟. We define two relations ≼^*₀ and ≼^*₁ on the set (^ωω)_Fin of finite-to-one functions which are Tukey equivalent to the eventual dominance relation of functions such that if ℱ⊆(^ωω)_Fin is ≼^*₁-unbounded, well-ordered by ≼^*₁, and not ≼^*₀-dominating, then there is a nonmeager p-ideal. The existence of such a system ℱ follows from Martin’s axiom. This is an analogue of the results of [3], [9,10] for increasing functions.