Quotients of Strongly Proper Forcings and Guessing Models

Cox, Sean; Krueger, John

doi:10.1017/jsl.2015.46

Cited by 14 publications

(56 citation statements)

References 12 publications

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“…3 2 Since we would like our forcings to be in H(κ ++ ), we will implicitly assume that Aa p(β) and fa p(β) are nice names, and ap(β) is a canonical name for their pair. 3 Let p = (ap, Xp) be a condition in Pα. We will often find it convenient to write ap(β) or Xp(γ) for β < α and γ ≤ α without necessarily knowing that β ∈ dom(ap) or γ ∈ dom(Xp).…”

Section: The Forcing Iterationmentioning

confidence: 99%

Club isomorphisms on higher Aronszajn trees

Krueger

2018

Annals of Pure and Applied Logic

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We prove the consistency, assuming an ineffable cardinal, of the statement that CH holds and any two normal countably closed ω 2 -Aronszajn trees are club isomorphic. This work generalizes to higher cardinals the property of Abraham-Shelah [1] that any two normal ω 1 -Aronszajn trees are club isomorphic, which follows from PFA. The statement that any two normal countably closed ω 2 -Aronszajn trees are club isomorphic implies that there are no ω 2 -Suslin trees, so our proof also expands on the method of Laver-Shelah [5] for obtaining the ω 2 -Suslin hypothesis.

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Section: The Forcing Iterationmentioning

confidence: 99%

Club isomorphisms on higher Aronszajn trees

Krueger

2018

Annals of Pure and Applied Logic

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“…We comment that in the model of [1,Section 7], GMP holds but there exists an ω 1 -Suslin tree. Since an ω 1 -Suslin tree is an example of a c.c.c., ω 1 -distributive forcing of size ω 1 , this provides a different proof that Todorčević's maximality principle does not follow from GMP.…”

Section: A Variation Of Easton's Lemmamentioning

confidence: 99%

“…The main point is that almost all of the known consequences of the existence of ω 1 -guessing models follow from the existence of weakly guessing models. 1 It is straightforward to check that the following three consequences of the existence of ω 1 -guessing models follow from the existence of weakly ω 1 -guessing models, by slight modifications of the proofs given in [2]. Proposition 3.13.…”

Section: Weak Approximation and Guessingmentioning

confidence: 99%

Namba Forcing, Weak Approximation, and Guessing

Cox¹,

Krueger²

2018

J. symb. log.

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We prove a variation of Easton's lemma for strongly proper forcings, and use it to prove that, unlike the stronger principle IGMP, GMP together with 2 ω ≤ ω 2 is consistent with the existence of an ω 1 -distributive nowhere c.c.c. forcing poset of size ω 1 . We introduce the idea of a weakly guessing model, and prove that many of the strong consequences of the principle GMP follow from the existence of stationarily many weakly guessing models. Using Namba forcing, we construct a model in which there are stationarily many indestructibly weakly guessing models which have a bounded countable subset not covered by any countable set in the model. Weiss[15] introduced the combinatorial principle ISP(κ), which characterizes supercompactness in the case that κ is inaccessible, but is also consistent for small values of κ such as ω 2 . The principle ISP(ω 2 ) follows from PFA, and ISP(ω 2 ) implies some of the strong consequences of PFA, such as the failure of the square principle at all uncountable cardinals. Viale-Weiss [14] introduced the idea of an ω 1 -guessing model, and proved that ISP(ω 2 ) is equivalent to the existence of stationarily many ω 1 -guessing models in P ω2 (H(θ)), for all cardinals θ ≥ ω 2 .Viale [13] proved that the singular cardinal hypothesis (SCH) follows from the existence of stationarily many ω 1 -guessing models which are also internally unbounded, which means that any countable subset of the model is covered by a countable set in the model. This raises the question of whether ISP(ω 2 ) alone implies SCH. A closely related question of Viale [13, Remark 4.3] is whether it is consistent to have ω 1 -guessing models which are not internally unbounded. Much of the work in this paper was motivated by these two questions.In this paper we introduce a weak form of ω 1 -guessing. Let κ be a regular uncountable cardinal. A model N of size ω 1 with κ ∈ N is said to be weakly κguessing if whenever f : sup(N ∩ κ) → On is a function such that for cofinally many α < sup(N ∩ κ), f ↾ α ∈ N , then there is a function g ∈ N with domain κ such that g ↾ sup(N ∩ κ) = f . We say that N is weakly guessing if N is weakly κ-guessing for all regular uncountable cardinals κ ∈ N . We will show that the existence of stationarily many weakly guessing models suffices to prove most of the strong consequences of ISP(ω 2 ), including the failure of square principles.

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“…Let us make an additional observation about the model constructed in [5]. Recall that the pseudo-splitting number p is the least size of a collection X of infinite subsets of ω, closed under finite intersections, for which there is no set b such that b \ a is finite for all a ∈ X. Viale [9,Lemma 4.2] proved that under the assumption that ω 1 < p, if χ ≥ ω 2 is a regular cardinal, N ∈ P ω2 (H(χ)), N ≺ H(χ), and N is ω 1 -guessing, then N is internally unbounded.…”

Section: Guessing Models and Gmpmentioning

confidence: 99%

“…In [10] it was asked whether ISP(ω 2 ) determines the value of the continuum. We answered this question negatively in [5], by showing that ISP(ω 2 ) is consistent with 2 ω having any value of uncountable cofinality greater than ω 1 .…”

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confidence: 97%

Indestructible guessing models and the continuum

Cox

Krueger

2017

Fund. Math.

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We introduce a stronger version of an ω 1 -guessing model, which we call an indestructibly ω 1 -guessing model. The principle IGMP states that there are stationarily many indestructibly ω 1 -guessing models. This principle, which follows from PFA, captures many of the consequences of PFA, including the Suslin hypothesis and the singular cardinal hypothesis. We prove that IGMP is consistent with the continuum being arbitrarily large.Proof. See [5, Lemma 1.6] Lemma 1.2. Let P be a regular suborder of Q. Then for all q ∈ Q, there is s ∈ P such that for all t ≤ s in P, q and t are compatible in Q. Moreover, this property of s is equivalent to s forcing in P that q is in Q/Ġ P .Proof. See [5, Lemmas 1.1, 1.3].Lemma 1.3. Let P and Q be forcing posets, and assume that P is a regular suborder of Q. If D is a dense subset of Q, then P forces that D ∩ (Q/Ġ P ) is a dense subset of Q/Ġ P .Proof. See [5, Lemma 1.5].Lemma 1.4. Let P and Q be forcing posets, and assume that P is a regular suborder of Q. Let G be a V -generic filter on P. Suppose that s ∈ G and p ∈ Q/G. Then s and p are compatible in Q/G.

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Quotients of Strongly Proper Forcings and Guessing Models

Cited by 14 publications

References 12 publications

Club isomorphisms on higher Aronszajn trees

Club isomorphisms on higher Aronszajn trees

Namba Forcing, Weak Approximation, and Guessing

Indestructible guessing models and the continuum

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