On tree ideals

  title={On tree ideals},
  author={Martin Goldstern and Miroslav Repick{\'y} and Saharon Shelah and Otmar Spinas},
Let 10 and m0 be the ideals associated with Laver and Miller forcing, respectively. We show that add(l0) < cov(10) and add(mO) < cov(mO) are consistent. We also show that both Laver and Miller forcing collapse the continuum to a cardinal < [ . INTRODUCTION AND NOTATION In this paper we investigate the ideals connected with the classical tree forcings introduced by Laver [La] and Miller [Mi]. Laver forcing L is the set of all trees p on <'co such that p has a stem and whenever s E p extends stem… 
Generic trees
  • O. Spinas
  • Mathematics
    Journal of Symbolic Logic
  • 1995
Abstract We continue the investigation of the Laver ideal ℓ0 and Miller ideal m0 started in [GJSp] and [GRShSp]; these are the ideals on the Baire space associated with Laver forcing and Miller
Different cofinalities of tree ideals
We introduce a general framework of generalized tree forcings, GTF for short, that includes the classical tree forcings like Sacks, Silver, Laver or Miller forcing. Using this concept we study the
The n-dimensional Laver and Miller ideals
We investigate the ideals associated with finite powers of Laver forcing and Miller forcing and show that among them only the two-dimensional ideal J(M2) is a or-ideal. By a forcing iteration of M2
If Q is a collection of trees, e.g. an arboreal forcing condition like in [3], then meaning of ⋃ Q is cleared. Formally, trees are contained in SeqX (finite sequences of elements from X) and any tree
Families of sets with nonmeasurable unions with respect to ideals defined by trees
Subfamilies of the ideal s0 introduced by Marczewski-Szpilrajn and ideals sp0, l0 analogously defined using complete Laver trees and Laver Trees respectively are considered and it is relatively consistent with ZFC that there exists a maximal almost disjoint family in the Baire space such that A is sp-nonmeasurable.
Strongly unbounded and strongly dominating sets generalized
We generalize the notions of unbounded and strongly dominating subset of the Baire space. We compare the corresponding ideals and tree ideals, in particular we find a condition which implies that
The distributivity numbers of finite products of P(ω)/fin
Generalizing [ShSp], for every n < ω we construct a ZFC-model where the distributivity number of r.o.(P(ω)/fin), h(n + 1), is smaller than the one of r.o.(P(ω)/fin). This answers an old problem of
Nonmeasurable sets and unions with respect to selected ideals especially ideals defined by trees
In this paper we consider nonmeasurablity with respect to sigma-ideals defined be trees. First classical example of such ideal is Marczewski ideal s_0. We will consider also ideal l_0 defined by
Abstract In this paper, we consider a notion of nonmeasurablity with respect to Marczewski and Marczewski-like tree ideals $s_0$ , $m_0$ , $l_0$ , $cl_0$ , $h_0,$ and $ch_0$ . We show


Towers on trees
We show that (under MA) for any c many dense sets in Laver forcing L there exists a a-centered Q C L such that all the given dense sets are dense in Q. In particular, MA implies that L satisfies MA
The Kunen-Miller Chart (Lebesgue Measure, the Baire Property, Laver Reals and Preservation Theorems for Forcing)
A decade-old problem of J. Baumgartner is answered and the last three open questions of the Kunen-Miller chart about measure and category are answered.
On the consistency of Borel's conjecture
For X a subset of [0, 1], there is a family of properties which X might have, each of which is stronger than X having Lebesgue measure zero, and each of which is trivially satisfied if X is
On a notion of smallness for subsets of the Baire space
Let us call a set A ⊆ ω^ω of functions from ω into ω σ-bounded if there is a countable sequence of functions (α_n: n Є ω)⊆ ω^ω such that every member of A is pointwise dominated by an element of that
Sacks forcing, Laver forcing, and Martin's axiom
It is given a proof that it is consistent that Sacks forcing collapses cardinals, and it is shown that Laver forcing does not collapse cardinals.
The Cichoń diagram
This work concludes the discussion of additivity, Baire number, uniformity, and covering for measure and category by constructing the remaining 5 models of Cichon's diagram.
On completely Ramsey sets
Cichoñ's diagram
  • J. Symbolic Logic
  • 1993