628 research outputs found

    A Generalization of Martin's Axiom

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    We define the ℵ1.5\aleph_{1.5} chain condition. The corresponding forcing axiom is a generalization of Martin's Axiom and implies certain uniform failures of club--guessing on ω1\omega_1 that don't seem to have been considered in the literature before.Comment: 36 page

    A five element basis for the uncountable linear orders

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    In this paper I will show that it is relatively consistent with the usual axioms of mathematics (ZFC) together with a strong form of the axiom of infinity (the existence of a supercompact cardinal) that the class of uncountable linear orders has a five element basis. In fact such a basis follows from the Proper Forcing Axiom, a strong form of the Baire Category Theorem. The elements are X, omega_1, omega_1^*, C, C^* where X is any suborder of the reals of cardinality aleph_1 and C is any Countryman line. This confirms a longstanding conjecture of Shelah.Comment: 21 page

    Forcing consequences of PFA together with the continuum large

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    We develop a new method for building forcing iterations with symmetric systems of structures as side conditions. Using our method we prove that the forcing axiom for the class of all the small finitely proper posets is compatible with a large continuum.Comment: 35 page

    Dependent choice, properness, and generic absoluteness

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    We show that Dependent Choice is a sufficient choice principle for developing the basic theory of proper forcing, and for deriving generic absoluteness for the Chang model in the presence of large cardinals, even with respect to -preserving symmetric submodels of forcing extensions. Hence, not only provides the right framework for developing classical analysis, but is also the right base theory over which to safeguard truth in analysis from the independence phenomenon in the presence of large cardinals. We also investigate some basic consequences of the Proper Forcing Axiom in, and formulate a natural question about the generic absoluteness of the Proper Forcing Axiom in and. Our results confirm as a natural foundation for a significant portion of classical mathematics and provide support to the idea of this theory being also a natural foundation for a large part of set theory

    Absoluteness via Resurrection

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    The resurrection axioms are forcing axioms introduced recently by Hamkins and Johnstone, developing on ideas of Chalons and Velickovi\'c. We introduce a stronger form of resurrection axioms (the \emph{iterated} resurrection axioms RAα(Γ)\textrm{RA}_\alpha(\Gamma) for a class of forcings Γ\Gamma and a given ordinal α\alpha), and show that RAω(Γ)\textrm{RA}_\omega(\Gamma) implies generic absoluteness for the first-order theory of Hγ+H_{\gamma^+} with respect to forcings in Γ\Gamma preserving the axiom, where γ=γΓ\gamma=\gamma_\Gamma is a cardinal which depends on Γ\Gamma (γΓ=ω1\gamma_\Gamma=\omega_1 if Γ\Gamma is any among the classes of countably closed, proper, semiproper, stationary set preserving forcings). We also prove that the consistency strength of these axioms is below that of a Mahlo cardinal for most forcing classes, and below that of a stationary limit of supercompact cardinals for the class of stationary set preserving posets. Moreover we outline that simultaneous generic absoluteness for Hγ0+H_{\gamma_0^+} with respect to Γ0\Gamma_0 and for Hγ1+H_{\gamma_1^+} with respect to Γ1\Gamma_1 with γ0=γΓ0≠γΓ1=γ1\gamma_0=\gamma_{\Gamma_0}\neq\gamma_{\Gamma_1}=\gamma_1 is in principle possible, and we present several natural models of the Morse Kelley set theory where this phenomenon occurs (even for all HγH_\gamma simultaneously). Finally, we compare the iterated resurrection axioms (and the generic absoluteness results we can draw from them) with a variety of other forcing axioms, and also with the generic absoluteness results by Woodin and the second author.Comment: 34 page
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