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Incomparable ω 1 ‐like models of set theory
Author(s) -
Fuchs Gunter,
Gitman Victoria,
Hamkins Joel David
Publication year - 2017
Publication title -
mathematical logic quarterly
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.473
H-Index - 28
eISSN - 1521-3870
pISSN - 0942-5616
DOI - 10.1002/malq.201500002
Subject(s) - uncountable set , mathematics , countable set , transitive relation , embedding , consistency (knowledge bases) , set theory , model theory , universe , discrete mathematics , set (abstract data type) , pairwise comparison , continuum hypothesis , combinatorics , computer science , artificial intelligence , statistics , physics , astrophysics , programming language , mathematical analysis
We show that the analogues of the embedding theorems of [3], proved for the countable models of set theory, do not hold when extended to the uncountable realm of ω 1 ‐like models of set theory. Specifically, under the ⋄ hypothesis and suitable consistency assumptions, we show that there is a family of 2 ω 1many ω 1 ‐like models of ZFC , all with the same ordinals, that are pairwise incomparable under embeddability; there can be a transitive ω 1 ‐like model of ZFC that does not embed into its own constructible universe; and there can be an ω 1 ‐like model of PA whose structure of hereditarily finite sets is not universal for the ω 1 ‐like models of set theory.