Silver trees and Cohen reals

Uncountable Trees and Cohen -Reals

“…When dealing with trees on ω, even if the most natural versions of amoeba usually do not have pure decision, some refinements can be defined in order to even get the Laver property. This is indeed possible for Sacks, Miller, Laver and Mathias forcing, whereas in [15] Spinas has shown this cannot be done for Silver forcing. Rather surprisingly, we show that the situation with trees on κ > ω is completely different, and we are going to show that pure decision gets very often lost.…”

“…The importance of such a topic is that it has crucial applications in questions concerning cardinal invariants associated with tree-ideals and regularity properties. In the standard case, such a topic has been extensively studied; see [12], [14], [1] and [8], [15] for important results in the context of 2 ω and ω ω . When dealing with trees on ω, even if the most natural versions of amoeba usually do not have pure decision, some refinements can be defined in order to even get the Laver property.…”

No Tukey reduction of Lebesgue null to Silver null sets

“…However, in [17] I showed M ≤ T I(Si), hence add(I(Si)) ≤ add(M) holds in ZFC and therefore every Silver amoeba that is proper and can be iterated to produce a model for ℵ 1 < add(I(Si)) must add Cohen reals. Actually, the new element of [17] is the ZFC-inequality add(I(Si)) ≤ cov(M), as add(I(Si)) ≤ b had already been proved in [18]. Note that add(I(Sa)) ≤ b is also true by a result of Simon [15].…”

A Sacks amoeba preserving distributivity of $\mathcal {P}(\omega )/\mathrm {fin}$

“…Let v 0 denote J(Si ), the ideal associated to Silver forcing Si (see Section 2.2). In [Spi16] the first author proved that the meager ideal M is Tukey reducible to v 0 , and hence add(v 0 ) ≤ add(M). When this paper was written in 2018 we conjectured that h < add(v 0 ) holds in the first author's model for cov(N ) < add(v 0 ) (see [Spi18]).…”

“…In [Spi16] the first author has shown that the meager ideal M is Tukey reducible to v 0 and hence add(v 0 ) ≤ add(M) holds in ZFC. Note that, as a consequence, the model for our Main Theorem 1.1 shows the consistency of add(v 0 ) < add(s 0 ).…”