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SUBCOMPLETE FORCING, TREES, AND GENERIC ABSOLUTENESS

Published online by Cambridge University Press:  23 October 2018

GUNTER FUCHS  and
KAETHE MINDEN
GUNTER FUCHS
Affiliation:
MATHEMATICS, THE GRADUATE CENTER OF THE CITY UNIVERSITY OF NEW YORK, 365 FIFTH AVENUE, NEW YORK, NY 10016, USA and MATHEMATICS, COLLEGE OF STATEN ISLAND OF CUNY, STATEN ISLAND, NY10314, USAE-mail: Gunter.Fuchs@csi.cuny.eduURL: http://www.math.csi.cuny/edu/∼fuchs
KAETHE MINDEN
Affiliation:
MATHEMATICS, MARLBORO COLLEGE, 2582 SOUTH ROAD, MARLBORO, VT05344, USA E-mail: kminden@marlboro.eduURL: https://kaetheminden.wordpress.com/
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Abstract

We investigate properties of trees of height ω1 and their preservation under subcomplete forcing. We show that subcomplete forcing cannot add a new branch to an ω1-tree. We introduce fragments of subcompleteness which are preserved by subcomplete forcing, and use these in order to show that certain strong forms of rigidity of Suslin trees are preserved by subcomplete forcing. Finally, we explore under what circumstances subcomplete forcing preserves Aronszajn trees of height and width ω1. We show that this is the case if CH fails, and if CH holds, then this is the case iff the bounded subcomplete forcing axiom holds. Finally, we explore the relationships between bounded forcing axioms, preservation of Aronszajn trees of height and width ω1 and generic absoluteness of ${\rm{\Sigma }}_1^1$-statements over first order structures of size ω1, also for other canonical classes of forcing.

Keywords

03E5003E5703E3503E5503E05subcomplete forcingforcing axiomsgeneric absolutenessSuslin treescontinuum hypothesis
Type
Articles
Information
The Journal of Symbolic Logic , Volume 83 , Issue 3 , September 2018 , pp. 1282 - 1305
Copyright
Copyright © The Association for Symbolic Logic 2018 

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