On cardinal characteristics of Yorioka ideals

Abstract

Yorioka introduced a class of ideals (parametrized by reals) on the Cantor space to prove that the relation between the size of the continuum and the cofinality of the strong measure zero ideal on the real line cannot be decided in sans-serifZFC. We construct a matrix iteration of c.c.c. posets to force that, for many ideals in that class, their associated cardinal invariants (i.e., additivity, covering, uniformity and cofinality) are pairwise different. In addition, we show that, consistently, the additivity and cofinality of Yorioka ideals does not coincide with the additivity and cofinality (respectively) of the ideal of Lebesgue measure zero subsets of the real line.

The covering number of the strong measure zero ideal can be above almost everything else

“…This is a natural generalization of preservation methods by Judah and Shelah [13] and Brendle [4]. Our presentation is closer to [7,Sect. 4].…”

“…It is known that add(N ) ≤ minadd ≤ add(M) and cof(M) ≤ supcof ≤ cof(N ) (see [7,19]), even more, it is not hard to see that minadd ≤ add(SN ). Yorioka's characterization of cof(SN ) is established as follows.…”

Matrix Iterations with Vertical Support Restrictions

“…Preservation theory. A generalization of the contents of this part, as well as complete proofs and more examples, can be found in [CM,Sect. 4].…”

“…as this statement is a conjunction of a Σ 1 1statement with a Π 1 1 -statement of the reals (see e.g [CM,. Claim 4.27]), it is also true in N .…”

On cardinal characteristics associated with the strong measure zero ideal

“…Figure 3 summarizes the relationship between the cardinal invariants associated with Yorioka ideals and the cardinals in Cichoń's diagram (see e.g. [9,14,6]). Here minnon = min {non(I f ) | f ∈ ω ω increasing}, supcov = sup {cov(I f ) | f ∈ ω ω increasing}, and minadd and supcof are defined analogously.…”