On Sacks forcing and the Sacks property

Geschke, Stefan; Quickert, Sandra

doi:10.1007/978-1-4020-2776-5_7

Cited by 15 publications

(9 citation statements)

References 15 publications

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“…Moreover, in this model cof(M) = ω 1 holds, see corollary 66 in [4]. But by 1.5 and 1.6 of [1] we always have cof(M) = ν ω , consequently in the Sacks model ν ω = ν ω 1 = ω 1 < c holds.…”

Section: Consequently π I [S] Is Indeed Nodecmentioning

confidence: 83%

Between countably compact and ω-bounded

Juhász

Soukup

Szentmiklóssy

2015

Topology and its Applications

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Section: Consequently π I [S] Is Indeed Nodecmentioning

confidence: 83%

Between countably compact and ω-bounded

Juhász

Soukup

Szentmiklóssy

2015

Topology and its Applications

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“…The same holds for random forcing and Lebesgue measurability. 6 The same holds for random forcing and Lebesgue measurability. For the proofs, see Lemma 2 (p. 367) in Woodin [20].…”

Section: The Following Are Equivalentmentioning

confidence: 87%

“…6 (For the definition of the uniformization property, see Definition 2.20.) Here we only assume the uniformization property for odd level projective classes because 1 2 does not have the uniformization property and these assumptions are true under suitable large cardinals assumptions or projective determinacy.…”

Section: The Following Are Equivalentmentioning

confidence: 99%

Projective absoluteness for Sacks forcing

Ikegami

2009

Arch. Math. Logic

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“…Note From now on, we will use (without proof) that Sacks forcing has the Sacks property (see Lemma 7.3.2 in [1]) and moreover, that the countable support iteration of proper forcings with the Sacks property also has the Sacks property. For more details of Sacks forcing and the Sacks property see [11].…”

Section: Sacks Indestructibilitymentioning

confidence: 99%

Ideals of independence

Fischer

Montoya

2019

Arch. Math. Logic

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We study two ideals which are naturally associated to independent families. The first of them, denoted J A , is characterized by a diagonalization property which allows along a cofinal sequence (the order type of which of uncountable cofinality) of stages along a finite support iteration to adjoin a maximal independent family. The second ideal, denoted id(A), originates in Shelah's proof of i < u in Shelah (Arch Math Log 31(6), 433-443, 1992). We show that for every independent family A, id(A) ⊆ J A and define a class of maximal independent families, to which we refer as densely maximal, for which the two ideals coincide. Building upon the techniques of Shelah (1992) we characterize Sacks indestructibility for such families in terms of properties of id(A) and devise a countably closed poset which adjoins a Sacks indestructible densely maximal independent family. Keywords Independent families • Sacks indestructibility • Constellations of cardinal characteristics Mathematics Subject Classification 03E17 • 03E35 The authors would like to thank the Austrian Science Fund (FWF) for the generous support through START Grant Y1012-N35.

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On Sacks forcing and the Sacks property

Cited by 15 publications

References 15 publications

Between countably compact and ω-bounded

Between countably compact and ω-bounded

Projective absoluteness for Sacks forcing

Ideals of independence

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