SUBCOMPLETE FORCING AND <i>ℒ</i>-FORCING

Jensen, Ronald Björn

doi:10.1142/9789814602648_0002

Cited by 27 publications

(53 citation statements)

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“…We begin by recalling the concept of subcompleteness of a partial order, as introduced by Jensen (see [10]). If M and N are models of the same first order language, then we write M ≺ N to express that M is an elementary submodel of N , and we write σ : M ≺ N to say that σ is an elementary embedding from M to N .…”

Section: Fragments Of Subcompleteness and Their Preservationmentioning

confidence: 99%

Subcomplete Forcing, Trees, and Generic Absoluteness

Fuchs¹,

Minden²

2018

J. symb. log.

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We investigate properties of trees of height ω 1 and their preservation under subcomplete forcing. We show that subcomplete forcing cannot add a new branch to an ω 1 -tree. We introduce fragments of subcompleteness which are preserved by subcomplete forcing, and use these in order to show that certain strong forms of rigidity of Suslin trees are preserved by subcomplete forcings. Finally, we explore under what circumstances subcomplete forcing preserves Aronszajn trees of height and width ω 1 . We show that this is the case if CH fails, and if CH holds, then this is the case iff the bounded subcomplete forcing axiom holds. Finally, we explore the relationships between bounded forcing axioms, preservation of Aronszajn trees of height and width ω 1 and generic absoluteness of Σ 1 1 -statements over first order structures of size ω 1 , also for other canonical classes of forcing.

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Section: Fragments Of Subcompleteness and Their Preservationmentioning

confidence: 99%

Subcomplete Forcing, Trees, and Generic Absoluteness

Fuchs¹,

Minden²

2018

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“…It is maybe a little surprising, then, that SCFA turns out to have many of the same consequences MM has. Thus, Jensen showed [Jen09a], [Jen14] In order to answer this question, we will establish a version for subcomplete forcing of a well-known result of Larson [Lar00, Theorem 4.3], asserting that Martin's Maximum is preserved by <ω 2 -directed closed forcing. The point is that Fact 3.1 admits the following abstract generalization:…”

Section: Weak Square Vs Reflectionmentioning

confidence: 99%

“…Proof. By [Jen09a], [Jen14], SCFA implies that 2 ℵ1 = ℵ 2 , 2 so that 2 ℵ0 < ℵ ω . As pointed out earlier, SCFA implies Refl(ω 1 , E µ ω ) for every regular cardinal µ > ω 1 .…”

Section: Proofmentioning

confidence: 99%

Weak square and stationary reflection

Fuchs

Rinot

2018

Acta Math. Hungar.

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It is well-known that the square principle λ entails the existence of a non-reflecting stationary subset of λ + , whereas the weak square principle * λ does not. Here we show that if µ cf(λ) < λ for all µ < λ, then * λ entails the existence of a non-reflecting stationary subset of E λ + cf(λ) in the forcing extension for adding a single Cohen subset of λ + .It follows that indestructible forms of simultaneous stationary reflection entail the failure of weak square. We demonstrate this by settling a question concerning the subcomplete forcing axiom (SCFA), proving that SCFA entails the failure of * λ for every singular cardinal λ of countable cofinality.

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“…So, by Fodor's theorem, there are a stationary

A_{0} \subseteq {(S_{ω}^{ω_{3}})}^{N}

, and an ordinal α 0 such that for all

α \in A_{0}

, we have

otp false(C_{α} false) = α_{0}

. Let C be generic for the forcing

P_{A_{0}}

to shoot a club of order type ω 1 through A 0 , which is subcomplete, by [, Lemma 6.3]. Let D be generic over

N [C]

for

Col (ω_{1}, ω_{3}^{N})

.…”

Section: Enhanced Bounded Forcing Axiomsmentioning

confidence: 99%

Subcomplete forcing principles and definable well‐orders

Fuchs

2018

Mathematical Logic Qtrly

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It is shown that the boldface maximality principle for subcomplete forcing, sans-serifMPsans-serifSCfalse(Hω2false), together with the assumption that the universe has only set many grounds, implies the existence of a well‐ordering of ℘(ω1) definable without parameters. The same conclusion follows from sans-serifMPsans-serifSCfalse(Hω2false), assuming there is no inner model with an inaccessible limit of measurable cardinals. Similarly, the bounded subcomplete forcing axiom, together with the assumption that x# does not exist, for some x⊆ω, implies the existence of a well‐ordering of ℘(ω1) which is Δ1‐definable without parameters, and normalΔ1false(Hω2false)‐definable using a subset of ω1 as a parameter. This well‐order is in L(℘false(ω1false)). Enhanced versions of bounded forcing axioms are introduced that are strong enough to have the implications of sans-serifMPsans-serifSCfalse(Hω2false) mentioned above, and along the way, a bounded forcing axiom for countably closed forcing is proposed.

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SUBCOMPLETE FORCING AND ℒ-FORCING

Cited by 27 publications

References 0 publications

Subcomplete Forcing, Trees, and Generic Absoluteness

Subcomplete Forcing, Trees, and Generic Absoluteness

Weak square and stationary reflection

Subcomplete forcing principles and definable well‐orders

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