Incomparable ω 1 ‐like models of set theory

Mathematical Logic Quarterly 63 (1-2):66-76 (2017)
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Abstract

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 ω1‐like models of set theory. Specifically, under the ⋄ hypothesis and suitable consistency assumptions, we show that there is a family of many ω1‐like models of, all with the same ordinals, that are pairwise incomparable under embeddability; there can be a transitive ω1‐like model of that does not embed into its own constructible universe; and there can be an ω1‐like model of whose structure of hereditarily finite sets is not universal for the ω1‐like models of set theory.

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Elementary embeddings and infinitary combinatorics.Kenneth Kunen - 1971 - Journal of Symbolic Logic 36 (3):407-413.
Blunt and topless end extensions of models of set theory.Matt Kaufmann - 1983 - Journal of Symbolic Logic 48 (4):1053-1073.
Recursively saturated $\omega_1$-like models of arithmetic.Roman Kossak - 1985 - Notre Dame Journal of Formal Logic 26 (4):413-422.

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