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Quotients of Strongly Proper Forcings and Guessing Models
[article]

2015
*
arXiv
*
pre-print

We prove that a wide class

arXiv:1406.3306v2
fatcat:jky47xnjpje27bb2pam7c6sbji
*of**strongly**proper**forcing*posets have*quotients*with strong properties. ... Specifically, we prove that*quotients**of**forcing*posets which have simple universal*strongly*generic conditions on a stationary set*of**models*by certain nice regular suborders satisfy the ω_1-approximation ...*Quotients**of**Strongly**Proper**Forcings*In this section we prove our main theorem on*quotients**of**strongly**proper**forcings*. ...##
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QUOTIENTS OF STRONGLY PROPER FORCINGS AND GUESSING MODELS

2016
*
Journal of Symbolic Logic (JSL)
*

We prove that a wide class

doi:10.1017/jsl.2015.46
fatcat:jmb7i4u5rnfp3luuavikvadg2q
*of**strongly**proper**forcing*posets have*quotients*with strong properties. ... Specifically, we prove that*quotients**of**forcing*posets which have universal*strongly*generic conditions on a stationary set*of**models*by certain nice regular suborders satisfy the ω 1-approximation property ...*Quotients**of**Strongly**Proper**Forcings*In this section we prove our main theorem on*quotients**of**strongly**proper**forcings*. ...##
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Indestructible Guessing Models and the Continuum
[article]

2019
*
arXiv
*
pre-print

We introduce a stronger version

arXiv:1510.05297v3
fatcat:f34xjkqcrfahdotmh5wc4tl7pi
*of*an ω_1-*guessing**model*, which we call an indestructibly ω_1-*guessing**model*. ... This principle, which follows from PFA, captures many*of*the consequences*of*PFA, including the Suslin hypothesis*and*the singular cardinal hypothesis. ... Suppose that Q is*strongly**proper*on a stationary set,*and*P is*proper*. Then P*forces*that Q is*strongly**proper*on a stationary set. Proof. Fix θ such that P*forces*that θ is a cardinal*and*θ ≥ λ Q . ...##
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Coding with ladders a well-ordering of the reals
[article]

2001
*
arXiv
*
pre-print

Any

arXiv:math/0104195v1
fatcat:yeadxlhzhzhohe6njqatwxhuq4
*model**of*ZFC + GCH has a generic extension (made with a poset*of*size aleph_2) in which the following hold: MA + 2^aleph_0= aleph_2+ there exists a Delta^2_1-well ordering*of*the reals. ... The proof consists in iterating posets designed to change at will the*guessing*properties*of*ladder systems on omega_1. Therefore, the study*of*such ladders is a main concern*of*this article. ... Given a*guessing*ladder η such that dom(η) ∩ range(η) = ∅ a*model**of*MA + 2 ℵ 0 = ℵ 2 can be obtained in which η is*strongly**guessing*. ...##
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Coding with ladders a well ordering of the reals

2002
*
Journal of Symbolic Logic (JSL)
*

change at will the

doi:10.2178/jsl/1190150099
fatcat:plneaw7xlnas7gsucwikcd3nve
*guessing*properties*of*ladder systems on ω 1. ... Any*model**of*ZFC + GCH has a generic extension (made with a poset*of*size ℵ2) in which the following hold: there exists a -well ordering*of*the reals, The proof consists in iterating posets designed to ... Given a*guessing*ladder η such that dom(η) ∩ range(η) = ∅ a*model**of*M A + 2 ℵ 0 = ℵ 2 can be obtained in which η is*strongly**guessing*. ...##
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JSL volume 81 issue 1 Cover and Front matter

2016
*
Journal of Symbolic Logic (JSL)
*

The BULLETIN, the JOURNAL

doi:10.1017/jsl.2016.2
fatcat:57zgexyc7fhshcs7f6ia4fz6w4
*and*the REVIEW*OF*SYMBOLIC LOGIC are the offi cial organs*of*the Association for Symbolic Logic, an international organization for furthering research in logic*and*the exchange ...*of*ideas among mathematicians, computer scientists, linguists,*and*others interested in this fi eld. ... chains in the Tukey structure*of*non-p-points, by NATASHA DOBRINEN .................................................... 237*Quotients**of**strongly**proper**forcings**and**guessing**models*, by SEAN COX*and*JOHN ...##
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On the consistency strength of the proper forcing axiom

2011
*
Advances in Mathematics
*

Using this, we argue to show that any

doi:10.1016/j.aim.2011.07.016
fatcat:jb4iwnxqevcazbeo7wzkvufmym
*of*the known methods for*forcing**models**of*PFA from a large cardinal assumption requires a*strongly*compact cardinal. ... We prove that the*proper**forcing*axiom PFA implies these principles hold for ω 2 . ... They are indebted to Menachem Magidor for supplying them with the idea*of*the proof*of*Theorem 6.9, that is, Claim 6.9.1. ...##
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Guessing models and the approachability ideal
[article]

2019
*
arXiv
*
pre-print

This principle implies ISP(ω_2)

arXiv:1802.10125v2
fatcat:c3nduoznjzcypc4fud5pvdzluu
*and*ISP(ω_3),*and*hence the tree property at ω_2*and*ω_3, the Singular Cardinal Hypothesis,*and*the failure*of*the weak square principle (ω_2,λ), for all regular λ≥ω_2. ... In addition, it implies that the restriction*of*the approachability ideal I[ω_2] to the set*of*ordinals*of*cofinality ω_1 is the non stationary ideal on this set. ... Hence N[G N ] is an ω 1 -*guessing**model*in V [G N ].On the other hand, by Lemma 4.7, the*quotient**forcing*Q N is C N st -*strongly**proper*,*and*C N st is stationary in P ω 1 (V λ ). ...##
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Forcing axioms, approachability, and stationary set reflection
[article]

2020
*
arXiv
*
pre-print

We also adapt some ideas

arXiv:1807.06129v3
fatcat:wo7ath4exfddbc6wswfght3axu
*of*Woodin to simplify*and*unify many arguments in the literature involving preservation*of**forcing*axioms. ... We prove that an implication*of*Fuchino-Usuba relating stationary reflection to a version*of*Strong Chang's Conjecture cannot be reversed; strengthen*and*simplify some results*of*Krueger about*forcing*...*Guessing**models**and**strongly**proper*club shooting. ...##
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Rado's conjecture and its Baire version
[article]

2019
*
arXiv
*
pre-print

In particular, we show a fragment

arXiv:1712.02455v4
fatcat:j64m432cu5egjfflmqk4dn6a7a
*of*PFA, that is the*forcing*axiom for Baire Indestructibly*proper**forcings*, is compatible with the Baire Rado's Conjecture. ... Then we discuss the strength*and*limitations*of*the Baire Rado's Conjecture regarding its interaction with simultaneous stationary reflection*and*some families*of*weak square principles. ... Note that the class*of*semi-*strongly**proper**forcings*here properly contains the class*of**strongly**proper**forcings*in the sense*of*Mitchell. ...##
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Joint Laver diamonds and grounded forcing axioms
[article]

2017
*
arXiv
*
pre-print

I explore two separate topics: the concept

arXiv:1705.04422v1
fatcat:cw7z423bcbby5ezi67jnz2mo7i
*of*jointness for set-theoretic*guessing*principles,*and*the notion*of*grounded*forcing*axioms. ... A family*of**guessing*sequences is said to be joint if all*of*its members can*guess*any given family*of*targets independently*and*simultaneously. ... for Martin's axiom, a*proper*ground*model*for the*proper**forcing*axiom etc. ...##
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SPM Bulletin 20
[article]

2007
*
arXiv
*
pre-print

; covering by translates

arXiv:math/0703894v1
fatcat:tynedt26pfcrjb7n2drjatvumu
*of*a compact nullsets;*and*a new Problem*of*the Issue. ... sets; almost disjoint families;*forcing*axioms; Splitting families; Sacks Reals; unconditionally closed*and*algebraic sets in groups; Stratifiability; Dissipated Compacta; partition relations; pi-bases ... These covering properties are also applied to investigate*models**of**strongly*compact cardinals or*of*strong*forcing*axioms like MM or PFA. http://arxiv.org/math/math.LO/0703091 Matteo Viale 2.9. ...##
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Implications for climate sensitivity from the response to individual forcings

2015
*
Nature Climate Change
*

Climate

doi:10.1038/nclimate2888
fatcat:wlv3x6rq55gdto5u777trtz47i
*models*predict a wide range for two commonly used definitions: the transient climate response (TCR: the warming after 70 years*of*CO 2 concentrations that rise at 1% per year),*and*the equilibrium ... When we use single*forcing*experiments to estimate these efficacies*and*calculate climate sensitivity from the observed twentieth-century warming, our estimates*of*both TCR*and*ECS are revised upward compared ... Acknowledgements Climate*modeling*at GISS is supported by the NASA*Modeling*, Analysis*and*Prediction program*and*resources supporting this work were provided by the NASA High-End Computing (HEC) Program ...##
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Isogeny in Superstable Groups
[article]

2011
*
arXiv
*
pre-print

We study

arXiv:1110.1766v1
fatcat:3625cvzapjgj3al4gmc5lg53v4
*and*develop a notion*of*isogeny for superstable groups. We prove several fundamental properties*of*the notion*and*then use it to formulate*and*prove uniqueness results. ... Connections to existing*model*theoretic notions are explained. ... The maps induced by φ G*and*φ K on the*quotient*H 2 /H 1 is an isogeny, since it is surjective onto its image*and*the kernel*of*the map is the*quotient**of*the kernel*of*an isogeny*and*τ (H 2 /H 1 ) = τ ...##
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Definable well-orders of H(ω 2) and GCH

2012
*
Journal of Symbolic Logic (JSL)
*

Assuming 2ℵ0 = ℵ1

doi:10.2178/jsl.7704030
fatcat:hglrqhdywjb3hm2czqa5jmiqmq
*and*2ℵ1 = ℵ2, we build a partial order that*forces*the existence*of*a well-order*of*H(ω 2) lightface definable over ⟨H(ω 1), ∈⟩*and*that preserves cardinal exponentiation*and*cofinalities ... in V [G] there is no coherent*strongly*type-*guessing*club-sequence with stationary domain disjoint from S,*and**of*height η. ... (A) uses Friedman's*forcing*from [F] for coding the ground*model*H(ω 2 ) (*and*a well-order*of*it) by a subset B*of*ω 1*and*(B) uses Asperó's*forcing*from [As2] to make B definable. ...
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