Forcing with quotients
英文摘要:
We study an extensive connection between factor forcings of Borel subsets of Polish spaces modulo a sigma-ideal, and factor forcings of subsets of countable sets modulo an ideal.
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Michael Hrusak, Jindrich Zapletal
We study an extensive connection between factor forcings of Borel subsets of Polish spaces modulo a sigma-ideal, and factor forcings of subsets of countable sets modulo an ideal.
机构、总结、中文摘要均由AI分析而来,可能存在错误,请以论文正文为准。
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We prove that a wide class of strongly proper forcing posets have quotients with strong properties. Specifically, we prove that quotients of forcing posets which have simple universal strongly generic conditions on a stationary set of models by certain nice regular suborders satisfy the $ω_1$-approximation property. We prove that the existence of stationarily many $ω_1$-guessing models in $P_{ω_2}(H(θ))$, for sufficiently large cardinals $θ$, is consistent with the continuum being arbitrarily large, solving a problem of Viale and Weiss.
Let mu be singular of uncountable cofinality. If mu>2^{cf(mu)}, we prove that in P=([mu]^mu,supseteq) as a forcing notion we have a natural complete embedding of Levy(aleph_0, mu^+) (so P collapses mu^+ to aleph_0) and even Levy(aleph_0, U_{J^{bd}_kappa}(mu)) . The ``natural'' means that the forcing ({p in [mu]^mu :p closed}, supseteq) is naturally embedded and is equivalent to the Levy algebra. If mu <2^{cf(mu)} we have weaker results.
The feeling that those two forcing notions-Cohen and Random-(equivalently the corresponding Boolean algebras Borel(R)/(meager sets), Borel(R)/(null sets)) are special, was probably old and widespread. A reasonable interpretation is to show them unique, or ``minimal'' or at least characteristic in a family of ``nice forcing'' like Borel. We shall interpret ``nice'' as Souslin as suggested by Judah Shelah [JdSh 292]. We divide the family of Souslin forcing to two, and expect that: among the first part, i.e. those adding some non-dominated real, Cohen is minimal (=is below every one), while among…
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A classical theorem of Hechler asserts that the structure $\left(ω^ω,\le^*\right)$ is universal in the sense that for any $σ$-directed poset P with no maximal element, there is a ccc forcing extension in which $\left(ω^ω,\le^*\right)$ contains a cofinal order-isomorphic copy of P. In this paper, we prove a consistency result concerning the universality of the higher analogue $\left(κ^κ,\le^S\right)$: Theorem. Assume GCH. For every regular uncountable cardinal $κ$, there is a cofinality-preserving GCH-preserving forcing extension in which for every analytic quasi-order Q over $κ^κ$ and every st…
Let αbe a countable ordinal and ¶(α) the collection of its subsets isomorphic to α. We show that the separative quotient of the set ¶(α) ordered by the inclusion is isomorphic to a forcing product of iterated reduced products of Boolean algebras of the form P(ω^γ)/I(ω^γ), where γis a limit ordinal or 1 and I(ω^γ) the corresponding ordinal ideal. Moreover, the poset ¶(α) is forcing equivalent to a two-step iteration P(ω)/Fin * π, where πis an ω_1-closed separative pre-order in each extension by P(ω)/Fin and, if the distributivity number is equal toω_1, to P(ω)/Fin. Also we analyze the quotients…
Product of Invariant Types Modulo Domination-Equivalence
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The Steprans forcing notion arises as a quotient of Borel sets modulo the ideal of $σ$-continuity of a certain Borel not $σ$-continuous function. We give a characterization of this forcing in the language of trees and using this characterization we establish such properties of the forcing as fusion and continuous reading of names. Although the latter property is usually implied by the fact that the associated ideal is generated by closed sets, we show it is not the case with Steprans forcing. We also establish a connection between Steprans forcing and Miller forcing thus giving a new descripti…
With every $σ$-ideal $I$ on a Polish space we associate the $σ$-ideal $I^*$ generated by the closed sets in $I$. We study the forcing notions of Borel sets modulo the respective $σ$-ideals $I$ and $I^*$ and find connections between their forcing properties. To this end, we associate to a $σ$-ideal on a Polish space an ideal on a countable set and show how forcing properties of the forcing depend on combinatorial properties of the ideal. For $σ$-ideals generated by closed sets we also study the degrees of reals added in the forcing extensions. Among corollaries of our results, we get necessary …
We develop a forcing poset with finite conditions which adds a partial square sequence on a given stationary set, with adequate sets of models as side conditions. We then develop a kind of side condition product forcing for simultaneously adding partial square sequences on multiple stationary sets. We show that certain quotients of such forcings have the $ω_1$-approximation property. We apply these ideas to prove, assuming the consistency of a greatly Mahlo cardinal, that it is consistent that the approachability ideal $I[ω_2]$ does not have a maximal set modulo clubs.
It is well known to generalize the meagre ideal replacing aleph_0 by a (regular) cardinal lambda > aleph_0 and requiring the ideal to be lambda^+-complete. But can we generalize the null ideal? In terms of forcing, this means finding a forcing notion similar to the random real forcing, replacing aleph_0 by lambda, so requiring it to be (<lambda)-complete. Of course, we would welcome additional properties generalizing the ones of the random real forcing. Returning to the ideal (instead forcing) we may look at the Boolean Algebra of lambda-Borel sets modulo the ideal. Surprisingly we get an posi…
Given a nonunital $\mathrm{C}^*$-algebra $A$ one constructs its corona algebra $\mathcal M(A)/A$. This is the noncommutative analog of the Čech-Stone remainder of a topological space. We analyze the two faces of these algebras: the first one is given assuming CH, and the other one arises when Forcing Axioms are assumed. In their first face, corona $\mathrm{C}^*$-algebras have a large group of automorphisms that includes nondefinable ones. The second face is the Forcing Axiom one; here the automorphism group of a corona $\mathrm{C}^*$-algebra is as rigid as possible, including only definable el…
Measurability with respect to ideals is tightly connected with absoluteness principles for certain forcing notions. We study a uniformization principle that postulates the existence of a uniformizing function on a large set, relative to a given ideal. We prove that for all $σ$-ideals $I$ such that the ideal forcing $\mathbb{P}_I$ of Borel sets modulo $I$ is proper, this uniformization principle is equivalent to an absoluteness principle for projective formulas with respect to $\mathbb{P}_I$ that we call internal absoluteness. In addition, we show that it is equivalent to measurability with res…
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本文通过力迫法为模态逻辑的扩张理论建立集合论翻译,证明其可靠性与完备性,并构建相应的Kripke模型。
We show that the separative quotient of the poset (P(L),\subset) of isomorphic suborders of a countable scattered linear order L is σ-closed and atomless. So, under the CH, all these posets are forcing-equivalent (to P(ω)/Fin).
A strong antidiamond principle (*c) is shown to be consistent with CH. This principle can be stated as a "P-ideal dichotomy": every P-ideal on omega-1 (i.e. an ideal that is sigma-directed under inclusion modulo finite) either has a closed unbounded subset of omega-1 locally inside of it, or else has a stationary subset of omega-1 orthogonal to it. We rely on Shelah's theory of parameterized properness for NNR iterations, and make a contribution to the theory with a method of constructing the properness parameter simultaneously with the iteration. Our handling of the application of the NNR ite…
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