Cofinalities of Marczewski-like ideals

Brendle, Joerg; Khomskii, Yurii; Wohofsky, Wolfgang

doi:10.4064/cm7113-5-2017

Cited by 9 publications

(5 citation statements)

References 3 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…By similar methods cof(v 0 ) > c (as well as cof(r 0 ) > c) can be shown. Finally, cof( 0 ) > c and cof(m 0 ) > c have been established by Brendle, Khomskii, and the second author [BKW17].…”

mentioning

confidence: 75%

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

Spinas

Wohofsky

2021

Fund. Math.

Self Cite

View full text Add to dashboard Cite

By iterating an increasing amoeba for Sacks forcing (implicitly introduced by Louveau, Shelah, and Veličković), we obtain a model in which h (i.e., the distributivity of P(ω)/fin) is smaller than the additivity of the Marczewski ideal (the ideal associated with Sacks forcing). The forcing is different from the usual amoeba for Sacks forcing: Unlike the latter, it has the pure decision and the Laver property, and therefore does not add Cohen reals. In our model, h < hω holds true, which answers a question by Repický who asked whether hω equals h in ZFC.

show abstract

mentioning

confidence: 75%

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

Spinas

Wohofsky

2021

Fund. Math.

Self Cite

View full text Add to dashboard Cite

show abstract

“…Its restriction to Borel sets equals the ideal of countable sets by the perfect set property for Borel sets. For Mathias forcing, τ P is the Ellentuck topology, and N P and I P are equal to the ideal of nowhere Ramsey sets (see [BKW18]). The ideal associated to Silver forcing consists of the completely doughnut null sets (see [Hal03]).…”

Section: 1mentioning

confidence: 99%

Lebesgue's Density Theorem and definable selectors for ideals

Müller,

Schlicht,

Schrittesser

et al. 2018

Preprint

View full text Add to dashboard Cite

We introduce a notion of density point and prove results analogous to Lebesgue's density theorem for various well-known ideals on Cantor space and Baire space. In fact, we isolate a class of ideals for which our results hold.In contrast to these results, we show that there is no reasonably definable selector that chooses representatives for the equivalence relation on the Borel sets of having countable symmetric difference. In other words, there is no notion of density which makes the ideal of countable sets satisfy an analogue to the density theorem.The proofs of the positive results use only elementary combinatorics of trees, while the negative results rely on forcing arguments.1 set B Ď P pωq of Borel codes and Σ 1 1 sets P, Q Ď P pωq ˆR such that ty P R | P px, yqu " ty P R | Qpx, yqu for all x P B and every Borel set is of this form. Write B x for this set, the Borel set coded by x. With such a coding, standard arguments on the complexity of measure (see [Kec73, Theorem 2.2.3]) show that D µ restricted to the Borel sets is induced by a definable map on the Borel codes as follows: There is a map D : B Ñ B such that B Dpxq " D µ pB x q

show abstract

“…The existence of such a base guarantees that an ideal has a cofinality not greater than c and so the existence of a universal set for the ideal is possible (although it may be a wild subset of ω ω × X). A good example of an ideal for which there is no universal set is the Marczewski ideal s 0 (let us recall that A ∈ s 0 , if for every perfect set P there is a perfect set Q ⊆ P for which Q ∩ A = ∅), since its cofinality is greater than c (see [8], also [4]).…”

Section: Introductionmentioning

confidence: 99%

Universal sets for ideals

Cieślak

Michalski

2018

Bull. Polish Acad. Sci. Math.

View full text Add to dashboard Cite

In this paper we consider a notion of universal sets for ideals. We show that there exist universal sets of minimal Borel complexity for classic ideals like null subsets of 2 ω and meager subsets of any Polish space, and demonstrate that the existence of such sets is helpful in establishing some facts about the real line in generic extensions. We also construct universal sets for E -the σ-ideal generated by closed null subsets of 2 ω , and for some ideals connected with forcing notions: K σ subsets of ω ω and the Laver ideal. We also consider Fubini products of ideals and show that there are Σ 0 3 universal sets for N ⊗M and M ⊗ N .X, M the σ-ideal of sets of the first category, N the σ-ideal of null subsets 2010 Mathematics Subject Classification: Primary 54H05; Secondary 03E57.

show abstract

Cofinalities of Marczewski-like ideals

Abstract: We show that the cofinalities of both the Miller ideal m 0 (the σ-ideal naturally related to Miller forcing M) and the Laver ideal ℓ 0 (related to Laver forcing L) are larger than the size of the continuum c in ZFC.

Cited by 9 publications

References 3 publications

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

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

Lebesgue's Density Theorem and definable selectors for ideals

Universal sets for ideals

Contact Info

Product

Resources

About