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ITERATION THEOREMS FOR SUBVERSIONS OF FORCING CLASSES

Part of: Set theory

Published online by Cambridge University Press:  17 March 2025

GUNTER FUCHS Open the ORCID record for GUNTER FUCHS [Opens in a new window]  and
COREY BACAL SWITZER Open the ORCID record for COREY BACAL SWITZER [Opens in a new window]
GUNTER FUCHS*
Affiliation:
DEPARTMENT OF MATHEMATICS THE GRADUATE CENTER OF THE CITY UNIVERSITY OF NEW YORK 365 FIFTH AVENUE, NEW YORK, NY 10016, USA AND DEPARTMENT OF MATHEMATICS, CUNY COLLEGE OF STATEN ISLAND 2800 VICTORY BOULEVARD, STATEN ISLAND NY 10314, USA URL: http://www.math.csi.cuny/edu/~fuchs
COREY BACAL SWITZER
Affiliation:
INSTITUTE OF MATHEMATICS, UNIVERSITY OF VIENNA KOLINGASSE 14-16, 1090 VIENNA AUSTRIA E-mail: corey.bacal.switzer@univie.ac.at
*
E-mail: Gunter.Fuchs@csi.cuny.edu
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Abstract

We prove various iteration theorems for forcing classes related to subproper and subcomplete forcing, introduced by Jensen. In the first part, we use revised countable support iterations, and show that 1) the class of subproper, ${}^\omega \omega $-bounding forcing notions, 2) the class of subproper, T-preserving forcing notions (where T is a fixed Souslin tree) and 3) the class of subproper, $[T]$-preserving forcing notions (where T is an $\omega _1$-tree) are iterable with revised countable support. In the second part, we adopt Miyamoto’s theory of nice iterations, rather than revised countable support. We show that this approach allows us to drop a technical condition in the definitions of subcompleteness and subproperness, still resulting in forcing classes that are iterable in this way, preserve $\omega _1$, and, in the case of subcompleteness, don’t add reals. Further, we show that the analogs of the iteration theorems proved in the first part for RCS iterations hold for nice iterations as well.

Keywords

iterated forcingrevised countable supportsubcomplete forcing

MSC classification

Primary: 03E50: Continuum hypothesis and Martin's axiom 03E55: Large cardinals 03E57: Generic absoluteness and forcing axioms 03E35: Consistency and independence results 03E17: Cardinal characteristics of the continuum
Secondary: 03E05: Other combinatorial set theory 03E40: Other aspects of forcing and Boolean-valued models

Information

Type
Article
Information
The Journal of Symbolic Logic , Volume 90 , Issue 1 , March 2025 , pp. 1 - 51
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Copyright
© The Author(s), 2025. Published by Cambridge University Press on behalf of The Association for Symbolic Logic

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References

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