20 research outputs found

    Class forcing, the forcing theorem and Boolean completions

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    The forcing theorem is the most fundamental result about set forcing, stating that the forcing relation for any set forcing is definable and that the truth lemma holds, that is everything that holds in a generic extension is forced by a condition in the relevant generic filter. We show that both the definability (and, in fact, even the amenability) of the forcing relation and the truth lemma can fail for class forcing. In addition to these negative results, we show that the forcing theorem is equivalent to the existence of a (certain kind of) Boolean completion, and we introduce a weak combinatorial property (approachability by projections) that implies the forcing theorem to hold. Finally, we show that unlike for set forcing, Boolean completions need not be unique for class forcing

    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

    Characterizations of pretameness and the Ord-cc

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    It is well known that pretameness implies the forcing theorem, and that pretameness is characterized by the preservation of the axioms of ZF−\mathsf{ZF}^-, that is ZF\mathsf{ZF} without the power set axiom, or equivalently, by the preservation of the axiom scheme of replacement, for class forcing over models of ZF\mathsf{ZF}. We show that pretameness in fact has various other characterizations, for instance in terms of the forcing theorem, the preservation of the axiom scheme of separation, the forcing equivalence of partial orders and their dense suborders, and the existence of nice names for sets of ordinals. These results show that pretameness is a strong dividing line between well and badly behaved notions of class forcing, and that it is exactly the right notion to consider in applications of class forcing. Furthermore, for most properties under consideration, we also present a corresponding characterization of the Ord\mathrm{Ord}-chain condition

    Forcing and the Universe of Sets: Must we lose insight?

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    A central area of current philosophical debate in the foundations of mathematics concerns whether or not there is a single, maximal, universe of set theory. Universists maintain that there is such a universe, while Multiversists argue that there are many universes, no one of which is ontologically privileged. Often forcing constructions that add subsets to models are cited as evidence in favour of the latter. This paper informs this debate by analysing ways the Universist might interpret this discourse that seems to necessitate the addition of subsets to VV. We argue that despite the prima facie incoherence of such talk for the Universist, she nonetheless has reason to try and provide interpretation of this discourse. We analyse extant interpretations of such talk, and argue that while tradeoffs in naturality have to be made, they are not too severe

    Forcing and the Universe of Sets: Must we lose insight?

    Get PDF
    A central area of current philosophical debate in the foundations of mathematics concerns whether or not there is a single, maximal, universe of set theory. Universists maintain that there is such a universe, while Multiversists argue that there are many universes, no one of which is ontologically privileged. Often forcing constructions that add subsets to models are cited as evidence in favour of the latter. This paper informs this debate by analysing ways the Universist might interpret this discourse that seems to necessitate the addition of subsets to VV. We argue that despite the prima facie incoherence of such talk for the Universist, she nonetheless has reason to try and provide interpretation of this discourse. We analyse extant interpretations of such talk, and argue that while tradeoffs in naturality have to be made, they are not too severe

    Forcing and the Universe of Sets: Must we lose insight?

    Get PDF
    A central area of current philosophical debate in the foundations of mathematics concerns whether or not there is a single, maximal, universe of set theory. Universists maintain that there is such a universe, while Multiversists argue that there are many universes, no one of which is ontologically privileged. Often forcing constructions that add subsets to models are cited as evidence in favour of the latter. This paper informs this debate by analysing ways the Universist might interpret this discourse that seems to necessitate the addition of subsets to VV. We argue that despite the prima facie incoherence of such talk for the Universist, she nonetheless has reason to try and provide interpretation of this discourse. We analyse extant interpretations of such talk, and argue that while tradeoffs in naturality have to be made, they are not too severe
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