Measure, category and projective wellorders

Fischer, Vera; Friedman, Sy D.; Khomskii, Yurii

doi:10.4115/jla.2014.6.8

Cited by 6 publications

(11 citation statements)

References 9 publications

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“…3 well-orders of the reals There has been significant interest in the study of possible constellations among the classical cardinal characteristics of the continuum in the presence of a projective, in fact ∆ 1 3 -definable, well-order of the reals (see [FF10,FFZ11,FFK14]). Answering a question of [FFK14], we show that each of the constellations described in the previous section is consistent with the existence of such a projective well-order. Since the proofs for the different constellations are very similar, we will only outline the proof of the following theorem.…”

Section: ∆mentioning

confidence: 99%

“…In [FF10] it is shown for example that various constellations involving a, b and s are consistent with the existence of a ∆ 1 3 well-order, while in [FFK14] it is shown that every admissible assignment of ℵ 1 and ℵ 2 to the characteristics in Cichoń's diagram is consistent with the existence of such a projective well-order. There is one main distinction between the various known methods for generically adjoining projective wellorders: methods relying on countable support S-proper iterations like in [FF10,FFK14], and methods using finite support iterations of ccc posets, e.g. [FFZ11,FFT12,FFZ13].…”

mentioning

confidence: 99%

“…In order to show that our new consistent constellations of Cichoń's diagram admit the existence of a ∆ 1 3 well-order of the reals, we further develop the second approach. Namely, we build up the method of almost disjoint coding which was introduced in [FFZ11] and in particular answer one of the open questions stated in [FFK14].…”

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confidence: 99%

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Coherent Systems of Finite Support Iterations

Fischer¹,

Friedman²,

Mejía³

et al. 2018

J. symb. log.

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We introduce a forcing technique to construct three-dimensional arrays of generic extensions through FS (finite support) iterations of ccc posets, which we refer to as 3D-coherent systems. We use them to produce models of new constellations in Cichoń's diagram, in particular, a model where the diagram can be separated into 7 different values. Furthermore, we show that this constellation of 7 values is consistent with the existence of a ∆ 1 3 well-order of the reals. IntroductionIn this paper, we provide a generalization of the method of matrix iteration, to which we refer as 3D-coherent systems of iterations and which can be considered a natural extension of the matrix method to include a third dimension. That is, if a matrix iteration can be considered as a system of partial orders P α,β : α ≤ γ, β ≤ δ such that whenever α ≤ α and β ≤ β then P α,β is a complete suborder of P α ,β , then our 3D-coherent systems are systems of posets P α,β,ξ :As an application of this method, we construct models where Cichoń's diagram is separated into different values, one of them with 7 different values. Moreover, these models determine the value of a, which is actually the same as the value of b, and we further show that such models can be produced so that they satisfy, additionally, the existence of a ∆ 1 3 well-order of the reals. The method of matrix iterations, or 2D-coherent systems of iterations in our terminology, has already a long history. It was introduced by Blass and Shelah in [BS89], to show that consistently u < d, where u is the ultrafilter number and d is the dominating number. The method was further developed in [BF11], where the terminology matrix iteration appeared for the first time, to show that if κ < λ are arbitrary regular uncountable cardinals then there is a generic extension in which a = b = κ < s = λ. Here a, b and s denote the almost disjointness, bounding and splitting numbers respectively. In [BF11], the authors also introduce a new method for the preservation of a mad (maximal almost disjoint) family along a matrix iteration, specifically a mad family added by H κ (Hechler's poset for adding a mad family, see Definition 4.1), a method which is of particular importance for our current work. Later, classical preservation properties for matrix iterations were improved by Mejía [Mej13a] to provide several examples of models where the cardinals in Cichoń's diagram assume many different values, in particular, a model with 6 different values. Since then, the question of how many distinct values there can be simultaneously in Cichoń's diagram has been of interest for many authors, see for example [FGKS] (a 2010 Mathematics Subject Classification. 03E17, 03E15, 03E35, 03E40, 03E45.

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Section: ∆mentioning

confidence: 99%

mentioning

confidence: 99%

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Coherent Systems of Finite Support Iterations

Fischer¹,

Friedman²,

Mejía³

et al. 2018

J. symb. log.

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“…The results for classes Σ 1 2 , Σ 1 2 are obtained by an elementary argument. On these and other theorems on uniformization and related questions see references above, as well as [37,34,10,4,7,6] with respect to modern studies, and also in the introductory section of our paper [16].…”

Section: Introductionmentioning

confidence: 92%

“…By definition, the multitree R ′ = ϕ(k, m ′ ) is an m ′ -collage over S ↑ , and then m-collage, too, by Lemma 9.3(i), where m = m ′ + 1. 6 To prove the openness let T ∈ D. Then T ↑ ∈ D ↑ , S ∈ S, and S ↑ T ↑ . We cannot assert directly that S T. However the multitree S ′ = S↑ (|T| ∪ |S|) also belongs to S by Definition 9.1(III).…”

Section: Validation Of 123(b)mentioning

confidence: 99%

Non-uniformizable sets with countable cross-sections on a given level of the projective hierarchy

Kanovei

Lyubetsky

2019

Fund. Math.

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We present a model of set theory, in which, for a given n ≥ 2, there exists a non-ROD-uniformizable planar Π 1 n set, whose all vertical cross-sections are countable sets (and in fact Vitali classes), while all planar Σ 1 n sets with countable vertical cross-sections are ∆ 1 n+1 -uniformizable. Thus it is true in this model, that the ROD-uniformization principle for sets with countable cross-sections fails for a given projective level but holds on all lower levels.

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