Perfect sets and collapsing continuum

Miroslav Repický

Commentationes Mathematicae Universitatis Carolinae (2003)

  • Volume: 44, Issue: 2, page 315-327
  • ISSN: 0010-2628

Abstract

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Under Martin’s axiom, collapsing of the continuum by Sacks forcing 𝕊 is characterized by the additivity of Marczewski’s ideal (see [4]). We show that the same characterization holds true if 𝔡 = 𝔠 proving that under this hypothesis there are no small uncountable maximal antichains in 𝕊 . We also construct a partition of ω 2 into 𝔠 perfect sets which is a maximal antichain in 𝕊 and show that s 0 -sets are exactly (subsets of) selectors of maximal antichains of perfect sets.

How to cite

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Repický, Miroslav. "Perfect sets and collapsing continuum." Commentationes Mathematicae Universitatis Carolinae 44.2 (2003): 315-327. <http://eudml.org/doc/249179>.

@article{Repický2003,
abstract = {Under Martin’s axiom, collapsing of the continuum by Sacks forcing $\mathbb \{S\}$ is characterized by the additivity of Marczewski’s ideal (see [4]). We show that the same characterization holds true if $\mathfrak \{d\}=\mathfrak \{c\}$ proving that under this hypothesis there are no small uncountable maximal antichains in $\mathbb \{S\}$. We also construct a partition of $^\omega 2$ into $\mathfrak \{c\}$ perfect sets which is a maximal antichain in $\mathbb \{S\}$ and show that $s^0$-sets are exactly (subsets of) selectors of maximal antichains of perfect sets.},
author = {Repický, Miroslav},
journal = {Commentationes Mathematicae Universitatis Carolinae},
keywords = {Sacks forcing; Marczewski's ideal; cardinal invariants; Sacks forcing; Marczewski's ideal; cardinal invariants},
language = {eng},
number = {2},
pages = {315-327},
publisher = {Charles University in Prague, Faculty of Mathematics and Physics},
title = {Perfect sets and collapsing continuum},
url = {http://eudml.org/doc/249179},
volume = {44},
year = {2003},
}

TY - JOUR
AU - Repický, Miroslav
TI - Perfect sets and collapsing continuum
JO - Commentationes Mathematicae Universitatis Carolinae
PY - 2003
PB - Charles University in Prague, Faculty of Mathematics and Physics
VL - 44
IS - 2
SP - 315
EP - 327
AB - Under Martin’s axiom, collapsing of the continuum by Sacks forcing $\mathbb {S}$ is characterized by the additivity of Marczewski’s ideal (see [4]). We show that the same characterization holds true if $\mathfrak {d}=\mathfrak {c}$ proving that under this hypothesis there are no small uncountable maximal antichains in $\mathbb {S}$. We also construct a partition of $^\omega 2$ into $\mathfrak {c}$ perfect sets which is a maximal antichain in $\mathbb {S}$ and show that $s^0$-sets are exactly (subsets of) selectors of maximal antichains of perfect sets.
LA - eng
KW - Sacks forcing; Marczewski's ideal; cardinal invariants; Sacks forcing; Marczewski's ideal; cardinal invariants
UR - http://eudml.org/doc/249179
ER -

References

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