945 research outputs found
On weakly tight families
Using ideas from Shelah's recent proof that a completely separable maximal
almost disjoint family exists when , we construct a
weakly tight family under the hypothesis \s \leq \b < {\aleph}_{\omega}. The
case when \s < \b is handled in \ZFC and does not require \b <
{\aleph}_{\omega}, while an additional PCF type hypothesis, which holds when
\b < {\aleph}_{\omega} is used to treat the case \s = \b. The notion of a
weakly tight family is a natural weakening of the well studied notion of a
Cohen indestructible maximal almost disjoint family. It was introduced by
Hru{\v{s}}{\'a}k and Garc{\'{\i}}a Ferreira \cite{Hr1}, who applied it to the
Kat\'etov order on almost disjoint families
Projective maximal families of orthogonal measures with large continuum
We study maximal orthogonal families of Borel probability measures on
(abbreviated m.o. families) and show that there are generic
extensions of the constructible universe in which each of the following
holds:
(1) There is a -definable well order of the reals, there is a
-definable m.o. family, there are no -definable
m.o. families and (in fact any reasonable
value of will do).
(2) There is a -definable well order of the reals, there is a
-definable m.o. family, there are no -definable
m.o. families, and .Comment: 12 page
Note on nonmeasurable unions
In this note we consider an arbitrary families of sets of ideal
introduced by Marczewski-Szpilrajn. We show that in any uncountable Polish
space and under some combinatorial and set theoretical assumptions
(cov(s_0)=\c for example), that for any family \ca\subseteq s_0 with
\bigcup\ca =X, we can find a some subfamily \ca'\subseteq\ca such that the
union \bigcup\ca' is not -measurable. We have shown a consistency of the
cov(s_0)=\omega_1<\c and existence a partition of the size \ca\in
[s_0]^{\omega} of the real line \bbr, such that there exists a subfamily
\ca'\subseteq\ca for which \bigcup\ca' is -nonmeasurable. We also showed
that it is relatively consistent with ZFC theory that \omega_1<\c and
existence of m.a.d. family \ca such that \bigcup\ca is -nonmeasurable in
Cantor space or Baire space . The consistency of
and is proved also.Comment: 12 page
Definable maximal cofinitary groups of intermediate size
Using almost disjoint coding, we show that for each
consistently ,
where is witnessed by a maximal cofinitary
group.Comment: 22 page
Forcing indestructibility of MAD families
AbstractLet A⊆[ω]ω be a maximal almost disjoint family and assume P is a forcing notion. Say A is P-indestructible if A is still maximal in any P-generic extension. We investigate P-indestructibility for several classical forcing notions P. In particular, we provide a combinatorial characterization of P-indestructibility and, assuming a fragment of MA, we construct maximal almost disjoint families which are P-indestructible yet Q-destructible for several pairs of forcing notions (P,Q). We close with a detailed investigation of iterated Sacks indestructibility
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