# On tree ideals

@inproceedings{Goldstern1993OnTI, title={On tree ideals}, author={Martin Goldstern and Miroslav Repick{\'y} and Saharon Shelah and Otmar Spinas}, year={1993} }

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…

## 35 Citations

Generic trees

- MathematicsJournal 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

- Mathematics
- 2017

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

- Mathematics
- 2001

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…

ON IDEALS WHICH COULD BE ASSOCIATED TO A POSET OF TREES

- Mathematics
- 2011

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

- MathematicsArch. Math. Log.
- 2015

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

- Mathematics
- 2014

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

- Mathematics
- 1998

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

- Mathematics
- 2015

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…

NONMEASURABLE SETS AND UNIONS WITH RESPECT TO TREE IDEALS

- MathematicsThe Bulletin of Symbolic Logic
- 2020

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…

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