Iterations of Boolean algebras with measure

Creature forcing and five cardinal characteristics in Cichoń’s diagram

“…A (by now) classical series of theorems [1,4,[7][8][9][10][14][15][16][17][18] proves these (in)equalities in ZFC and shows that they are the only ones provable. More precisely, all assignments of the values ℵ 1 and ℵ 2 to the characteristics in Cichoń's Diagram are consistent, provided they do not contradict the above (in)equalities.…”

“…Definition 2.5.1 A compound creature c consists of (1) natural numbers m dn < m up ; (2) a nonempty, finite 9 subset supp of (3) for each ξ ∈ supp ∩ sk a Sacks column c(ξ ) between m dn and m up ; (4) for each ξ ∈ supp ∩ non-sk and each subatomic sublevel u = ( , j) with m dn ≤ < m up a subatom c(ξ, u) ∈ K ξ,u ; and (5) for each m dn ≤ < m up a real number d( ) ≥ 0, called the "halving parameter" of c at level ). 10 We additionally require "modesty": 11 (6) for each subatomic sublevel u with m dn < u < m up there is at most one ξ ∈ supp ∩ non-sk such that the subatom c(ξ, u) is not a singleton.…”

On properties of compacta that do not reflect in small continuous images

“…We shall work in the measure algebra A of the Lebesgue measure on [0, 1]; the corresponding measure on A is still denoted by λ. The following is an immediate consequence of a result due to Kamburelis [12], Lemma 3.1; see also [5], Theorem 4.4. We can now give a (partial) negative solution to Question 4.3 from [25].…”

Splitting, Bounding, and Almost Disjointness Can Be Quite Different

“…Judah and Shelah [JS90] proved that given an infinite cardinal ν < θ 0 , every ν-centered forcing notion is θ 0 -∈ * -good. Moreover, as a consequence of results of Kamburelis [Kam89], any subalgebra 5 of B is ∈ * -good. 4 [BaJ95, Def.…”