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Condensation-coherent global square systems

In Anil Nerode & Richard A. Shore (eds.), Recursion Theory. American Mathematical Society. pp. 42--237 (1985)

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  1. Organic and tight.J. Cummings, M. Foreman & E. Schimmerling - 2009 - Annals of Pure and Applied Logic 160 (1):22-32.
    We define organic sets and organically stationary sequences, which generalize tight sets and tightly stationary sequences respectively. We show that there are stationary many inorganic sets and stationary many sets that are organic but not tight. Working in the Constructible Universe, we give a characterization of organic and tight sets in terms of fine structure. We answer a related question posed in [J. Cummings, M. Foreman, M. Magidor, Canonical structure in the universe of set theory: Part two, Ann. Pure Appl. (...)
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  • Canonical structure in the universe of set theory: Part two.James Cummings, Matthew Foreman & Menachem Magidor - 2006 - Annals of Pure and Applied Logic 142 (1):55-75.
    We prove a number of consistency results complementary to the ZFC results from our paper [J. Cummings, M. Foreman, M. Magidor, Canonical structure in the universe of set theory: part one, Annals of Pure and Applied Logic 129 211–243]. We produce examples of non-tightly stationary mutually stationary sequences, sequences of cardinals on which every sequence of sets is mutually stationary, and mutually stationary sequences not concentrating on a fixed cofinality. We also give an alternative proof for the consistency of the (...)
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