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MITCHELL-INSPIRED FORCING, WITH SMALL WORKING PARTS AND COLLECTIONS OF MODELS OF UNIFORM SIZE AS SIDE CONDITIONS, AND GAP-ONE SIMPLIFIED MORASSES

Part of: Set theory

Published online by Cambridge University Press:  22 June 2020

CHARLES MORGAN
CHARLES MORGAN*
Affiliation:
DEPARTMENT OF MATHEMATICS UNIVERSITY COLLEGE LONDON GOWER STREET, LONDONWC1E 6BT, UKE-mail: charles.morgan@ucl.ac.uk
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Abstract

We show that a $(\kappa ^{+},1)$ -simplified morass can be added by a forcing with working parts of size smaller than $\kappa $ . This answers affirmatively the question, asked independently by Shelah and Velleman in the early 1990s, of whether it is possible to do so.

Our argument use a modification of a technique of Mitchell’s for adding objects of size $\omega _2$ in which collections of models – all of equal, countable size – are used as side conditions. In our modification, whilst the individual models are, as in Mitchell’s technique, taken ad hoc from quite general classes, the collections of models are very highly structured, in a way that is somewhat different from, perhaps more stringent than, Mitchell’s original, arguably making the method more wieldy and giving the prospect of further uses with more delicate working parts.

Keywords

forcingproper forcingside conditionssimplified morasses

MSC classification

Primary: 03E35: Consistency and independence results
Secondary: 03E05: Other combinatorial set theory 03E30: Axiomatics of classical set theory and its fragments 03E45: Inner models, including constructibility, ordinal definability, and core models
Type
Article
Information
The Journal of Symbolic Logic , Volume 87 , Issue 1 , March 2022 , pp. 392 - 415
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Copyright
© The Author(s), 2020. Published by Cambridge University Press on behalf of The Association for Symbolic Logic

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