Indivisible sets and well‐founded orientations of the Rado graph

Mathematical Logic Quarterly 65 (1):46-56 (2019)
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Every set can been thought of as a directed graph whose edge relation is ∈. We show that many natural examples of directed graphs of this kind are indivisible: for every infinite κ, for every indecomposable λ, and every countable model of set theory. All of the countable digraphs we consider are orientations of the countable random graph. In this way we find indivisible well‐founded orientations of the random graph that are distinct up to isomorphism, and ℵ1 that are distinct up to siblinghood.



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Forcing with copies of the Rado and Henson graphs.Osvaldo Guzmán & Stevo Todorcevic - 2023 - Annals of Pure and Applied Logic 174 (8):103286.

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Elementary embeddings and infinitary combinatorics.Kenneth Kunen - 1971 - Journal of Symbolic Logic 36 (3):407-413.
On Interpretations of Arithmetic and Set Theory.Richard Kaye & Tin Lok Wong - 2007 - Notre Dame Journal of Formal Logic 48 (4):497-510.

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