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1

Masas in the Calkin algebra without the Continuum Hypothesis

Shelah S., Steprans J.

Journal of Applied Analysis

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2011

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Vol. 17, nr 1

69-89

EN

Methods for constructing masas in the Calkin algebra without assuming the Continuum Hypothesis are developed.

2

Specializing Aronszajn tree and preserving some weak diamonds

Mildenberger H., Shelah S.

Journal of Applied Analysis

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2009

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Vol. 15, nr 1

47-78

EN

We show that [wzór) together with CH and "all Aronszajn trees are special" is consistent relative to ZFC. The weak diamond for the covering relation of Lebesgue null sets was the only weak diamond in the Cichori diagramrne for relations whose consistency together with "all Aronszajn trees are special" was not yet settled. Our forcing proof gives also new proofs to the known consistencies of several other weak diamonds stemming from the Cichori diagramme together with "all Aronszajn trees are special" and CH. The main part of our work is an application [15, Chapter V, §§ 1-7] for a special completeness system, such that we have a genericity game. Thus we show new preservation properties of the known forcings.

3

Non-Cohen oracle c.c.c.

Shelah S.

Journal of Applied Analysis

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2006

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Vol. 12, nr 1

1-17

EN

The oracle c.c.c. is closely related to Cohen forcing. During an iteration we can "omit a type" ; i.e. preserve "the intersection of a given family of Borel sets of reals is empty" provided that Cohen forcing satisfies it. We generalize this to other cases. In Section 1 we replace Cohen by "nicely" definable c.c.c., do the parallel of the oracle c.c.c. and end with a criterion for extracting a subforcing (not a complete iubforcing, <· ! ) of a given nicely one and satisfying the oracle.

4

On nicely definable forcing notions

Shelah S.

Journal of Applied Analysis

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2005

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Vol. 11, nr 1

1-17

EN

We prove that if Q is a new-nep forcing then it cannot add a dominating real. We also show that amoeba forcing cannot be P(X) / I if I is an N1-complete ideal. Furthermore, we generalize the results of [12].

5

Properness without elementaricity

Shelah S.

Journal of Applied Analysis

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2004

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Vol. 10, nr 2

169-289

EN

We present reasons for developing a theory of forcing notions which satisfy the properness demand for countable models which are not necessarily elementary sub-models of some (H(x),∈ ). This leads to forcing notions which are "reasonably" definable. We present two specific properties materializing this intuition: nep (non-elernentary properness) and snep (Souslin non-elementary properness) and also the older Souslin proper. For this we consider candidates (countable models to which the definition applies). A major theme here is "preservation by iteration", but we also show a dichotomy: if such forcing notions preserve the positiveness of the set of old reals for some naturally defined c.c.c. ideal, then they preserve the positiveness of any old positive set hence preservation by composition of two follows. Last but not least, we prove that (among such forcing notions) the only one commuting with Cohen is Cohen itself; in other words, any other such forcing notion make the set of old reals to a meager set. In the end we present some open problems in this area.

6

Critical cardinalities and additivity properties of combinatorial notions of smallness

Shelah S., Tsaban B.

Journal of Applied Analysis

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2003

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Vol. 9, nr 2

149-162

EN

Motivated by the minimal tower problem, an earlier work studied diagonalizations of covers where the covers are related to linear quasiorders (r-covers). We deal with two types of combinatorial questions which arise from this study. 1. Two new cardinals introduced in the topological study are expressed in terms of well known cardinals characteristics of the continuum. 2. We study the additivity numbers of the combinatorial notions corresponding to the topological diagonalization notions. This gives new insights on the structure of the eventual dominance ordering on the Baire space, the almost inclusion ordering on the Rothberger space, and the interactions between them.

7

Category analogue of sup-measurability problem

Ciesielski K., Shelah S.

Journal of Applied Analysis

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2000

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Vol. 6, nr 2

159-172

EN

A function F: R2→ R is called sup-measurable if Ff : R→ R given by Ff(x) = F(x,f(x)), x ∈R, is measurable for each measurable function f: R→ R. It is known that under different set theoretical assumptions, including CH, there are sup-measurable non-measurable functions, as well as their category analogues. In this paper we will show that the existence of the category analogues of sup-measurable non-measurable functions is independent of ZFC. A similar result for the original measurable case is the subject of a work in prepartion by Rosłanowski and Shelah.

8

The lifting problem with the full ideal

Shelah S.

Journal of Applied Analysis

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1998

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Vol. 4, nr 1

1-17

EN

We show that there are a cardinal μ, a σ-ideal I ⊆ P(μ) and a σ-subalgebra B of subsets of μ extending I such that B/I satisfies the c.c.c. but the quotient algebra B/I has no lifting.

9

A result related to the problem cn of Fremlin

Kolman O., Shelah S.

Journal of Applied Analysis

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1998

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Vol. 4, nr 2

161-165

EN

We show that the set of injective functions from any uncountable cardinal less than the continuum into the real numbers is of the second category in the box product topology.