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UNIVERSALITY PROPERTIES OF FORCING

Part of: Model theory Set theory

Published online by Cambridge University Press:  06 March 2025

FRANCESCO PARENTE Open the ORCID record for FRANCESCO PARENTE [Opens in a new window]  and
MATTEO VIALE
FRANCESCO PARENTE
Affiliation:
GRADUATE SCHOOL OF SYSTEM INFORMATICS KOBE UNIVERSITY 1-1 ROKKODAI-CHO NADA-KU, KOBE 657-8501 JAPAN E-mail: francesco.parente@people.kobe-u.ac.jp
MATTEO VIALE*
Affiliation:
DIPARTIMENTO DI MATEMATICA “GIUSEPPE PEANO” UNIVERSITÀ DI TORINO VIA CARLO ALBERTO 10, 10123 TURIN ITALY
*
E-mail: matteo.viale@unito.it
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Abstract

The purpose of this paper is to investigate forcing as a tool to construct universal models. In particular, we look at theories of initial segments of the universe and show that any model of a sufficiently rich fragment of those theories can be embedded into a model constructed by forcing. Our results rely on the model-theoretic properties of good ultrafilters, for which we provide a new existence proof on non-necessarily complete Boolean algebras.

Keywords

forcinguniversalitygood ultrafilter

MSC classification

Primary: 03E40: Other aspects of forcing and Boolean-valued models
Secondary: 03C50: Models with special properties (saturated, rigid, etc.) 03E05: Other combinatorial set theory

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Type
Article
Information
The Journal of Symbolic Logic , First View , pp. 1 - 19
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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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