Parameterized partition relations on the real numbers

Bagaria, Joan; Prisco, Carlos Augusto Di

doi:10.1007/s00153-009-0121-y

Cited by 5 publications

(9 citation statements)

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“…• Shoenfield's absoluteness theorem is a generic absoluteness result for Θ the family of Σ 1 2 -properties with real parameters, Γ the class of all forcings, T = ZFC. Further results pin down the exact large cardinal strength of the assertion that L(R) is generically invariant with respect to certain classes of forcings (among others see [4,5,6,7,26,27,28]).…”

Section: Introductionmentioning

confidence: 89%

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Absoluteness via resurrection

Audrito

Viale

2017

J. Math. Log.

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The resurrection axioms are forcing axioms introduced recently by Hamkins and Johnstone, developing on ideas of Chalons and Veličković. We introduce a stronger form of resurrection axioms (the iterated resurrection axioms RAα(Γ) for a class of forcings Γ and a given ordinal α), and show that RAω(Γ) implies generic absoluteness for the first-order theory of H γ + with respect to forcings in Γ preserving the axiom, where γ = γΓ is a cardinal which depends on Γ (γΓ = ω1 if Γ 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 H γ + 0 with respect to Γ0 and for H γ + 1 with respect to Γ1 with γ0 = γΓ 0 = γΓ 1 = γ1 is in principle possible, and we present several natural models of the Morse Kelley set theory where this phenomenon occurs (even for all Hγ 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.

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Section: Introductionmentioning

confidence: 89%

“…Let g : λ → λ and λ inaccessible be such that f = g ↾ κ and V λ+1 |= T . Since Γ ∩ V κ , Σ ∩ V κ is Γ, Σ as computed in V κ+1 and the same holds for λ, letting F = {P Γ,g α : α < λ}, we have that 7 We recall thatî β is the natural embedding from V P Γ,f β to V P Γ,f κ (see Proposition 2.9).…”

Section: Straigthforward Given Thatmentioning

confidence: 99%

Absoluteness via resurrection

Audrito

Viale

2017

J. Math. Log.

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“…There are other strong forms of the Ramsey property known to hold in the Solovay model. For example, Bagaria and Di Prisco [1] use generic-absoluteness results to establish parametrized partition properties, some of which generalize the Ramsey property. Some regularity properties of this kind (the Perfect Set Property, for instance) hold even in L(R)[U ], the extension of the Solovay model by a generic ultrafilter (see [7]), though in that model there are sets without the Ramsey property, non-Lebesgue-measurable sets, and sets without the Baire property.…”

Section: Lemma 27 An Infinite Almost-disjoint Family a Is A Mad Fammentioning

confidence: 99%

HAPPY AND MAD FAMILIES INL(ℝ)

Neeman

Norwood

2018

J. symb. log.

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We prove that, in the choiceless Solovay model, every set of reals isH-Ramsey for every happy familyHthat also belongs to the Solovay model. This gives a new proof of Törnquist’s recent theorem that there are no infinite mad families in the Solovay model. We also investigate happy families and mad families under determinacy, applying a generic absoluteness result to prove that there are no infinite mad families under$A{D^ + }$.

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“…In the case of (4), the theorem follows from Bagaria and Bosch [2]. And for P satisfying (5) or (6), it follows from the results of Bagaria and Bosch [1], and Bagaria and Di Prisco [3], respectively.…”

Section: Which Forcing Notions Don't Do Itmentioning

confidence: 85%

“…(3) ϑ is a weakly-compact cardinal in L, and P is ccc; (4) ϑ is a Mahlo cardinal in L, and P is σ-linked; (5) [3]), so, accordingly, this theorem includes the previous one.…”

Section: Which Forcing Notions Don't Do Itmentioning

confidence: 99%