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ARONSZAJN TREE PRESERVATION AND BOUNDED FORCING AXIOMS

Part of: Set theory

Published online by Cambridge University Press:  01 February 2021

GUNTER FUCHS
GUNTER FUCHS*
Affiliation:
DEPARTMENT OF MATHEMATICS, THE GRADUATE CENTER THE CITY UNIVERSITY OF NEW YORK 365 FIFTH AVENUE, NEW YORK, NY10016, USA DEPARTMENT OF MATHEMATICS, COLLEGE OF STATEN ISLAND THE CITY UNIVERSITY OF NEW YORK STATEN ISLAND, NEW YORK, NY10314, USAE-mail: [email protected]: http://www.math.csi.cuny/edu/~fuchs
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Abstract

I investigate the relationships between three hierarchies of reflection principles for a forcing class $\Gamma $ : the hierarchy of bounded forcing axioms, of $\Sigma ^1_1$ -absoluteness, and of Aronszajn tree preservation principles. The latter principle at level $\kappa $ says that whenever T is a tree of height $\omega _1$ and width $\kappa $ that does not have a branch of order type $\omega _1$ , and whenever ${\mathord {\mathbb P}}$ is a forcing notion in $\Gamma $ , then it is not the case that ${\mathord {\mathbb P}}$ forces that T has such a branch. $\Sigma ^1_1$ -absoluteness serves as an intermediary between these principles and the bounded forcing axioms. A special case of the main result is that for forcing classes that don’t add reals, the three principles at level $2^\omega $ are equivalent. Special attention is paid to certain subclasses of subcomplete forcing, since these are natural forcing classes that don’t add reals.

Keywords

subcomplete forcingforcing axiomsgeneric absolutenessAronszajn treescontinuum hypothesis

MSC classification

Primary: 03E50: Continuum hypothesis and Martin's axiom
Secondary: 03E57: Generic absoluteness and forcing axioms 03E35: Consistency and independence results 03E55: Large cardinals 03E05: Other combinatorial set theory
Type
Article
Information
The Journal of Symbolic Logic , Volume 86 , Issue 1 , March 2021 , pp. 293 - 315
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Copyright
© The Association for Symbolic Logic 2021

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