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THE ONTO MAPPING OF SIERPINSKI AND NONMEAGER SETS.
- Source :
- Journal of Symbolic Logic; Sep2017, Vol. 82 Issue 3, p958-965, 8p
- Publication Year :
- 2017
-
Abstract
- The principle ( ) of Sierpinski is the assertion that there is a family of functions $\left\{ {{\varphi _n}:{\omega _1} \to {\omega _1}|n \in \omega } \right\}$ such that for every $I \in {[{\omega _1}]^{{\omega _1}}}$ there is n ε ω such that ${\varphi _n}[I] = {\omega _1}$. We prove that this principle holds if there is a nonmeager set of size ω1 answering question of Arnold W. Miller. Combining our result with a theorem of Miller it then follows that ( ) is equivalent to $non\left( {\cal M} \right) = {\omega _1}$. Miller also proved that the principle of Sierpinki is equivalent to the existence of a weak version of a Luzin set, we will construct a model where all of these sets are meager yet $non\left( {\cal M} \right) = {\omega _1}$. [ABSTRACT FROM AUTHOR]
Details
- Language :
- English
- ISSN :
- 00224812
- Volume :
- 82
- Issue :
- 3
- Database :
- Complementary Index
- Journal :
- Journal of Symbolic Logic
- Publication Type :
- Academic Journal
- Accession number :
- 125072212
- Full Text :
- https://doi.org/10.1017/jsl.2016.24