Creature forcing and large continuum: the joy of halving

Abstract

Abstract. For f, g ∈ ω ω let c ∀ f ,g be the minimal number of uniform g-splitting trees needed to cover the uniform f -splitting tree, i.e., for every branch ν of the f -tree, one of the g-trees contains ν. Let c ∃ f ,g be the dual notion: For every branch ν, one of the g-trees guesses ν(m) infinitely often. We show that it is consistent that c ∃ fǫ ,gǫ = c ∀ fǫ ,gǫ = κ ǫ for continuum many pairwise different cardinals κ ǫ and suitable pairs ( f ǫ , g ǫ ). For the proof we introduce a new mixed-limit creature forcing construction.

Creature forcing and five cardinal characteristics in Cichoń’s diagram

“…• the width norm; 12 As usual, if the logarithm results in a negative number, or if we apply the logarithm to a negative number, we instead define the resulting norm to be 0. So really we mean nor…”

On cardinal characteristics of Yorioka ideals

“…Other way to attack Question 6.1 is by large products of creature forcing (cf. [9,20,21]). Quite recently, A. Fischer, Goldstern, Kellner and Shelah [9] used this technique to prove that 5 cardinal invariants on the right of Cichoń's diagram are pairwise different.…”

“…Goldstern, Kellner, and Shelah have studied cardinal coefficients of the form and for (with a different notation in their work) to provide the first examples of models where continuum many cardinal characteristics of the continuum are pairwise different . On the other hand, Brendle and Mejía investigated cardinals of the form for in relation with gap numbers for F σ ‐ideals .…”

“…The cardinal is known from Miller's characterization (for any ) . On the other hand, Kellner and Shelah constructed a proper poset that forces that, for the types and , continuum many cardinals of each type are pairwise different . In fact, Kamo and Osuga discovered a connection between the cardinals of type , and the anti‐localization cardinals (Lemmas & ; ), which they use to construct a finite support iteration of c.c.c.…”

Nonproper products

“…It occurs that a strong version of halving from [4, Section 2.2] implies the properness of forcing notions of the type Q * ∞ (K, Σ). More on applications of halving can be found in Kellner and Shelah [2,3] and Ros lanowski and Shelah [5].…”