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Quotients of Strongly Proper Forcings and Guessing Models [article]

Sean Cox, John Krueger
2015 arXiv   pre-print
We prove that a wide class 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.  ... 
arXiv:1406.3306v2 fatcat:jky47xnjpje27bb2pam7c6sbji

QUOTIENTS OF STRONGLY PROPER FORCINGS AND GUESSING MODELS

SEAN COX, JOHN KRUEGER
2016 Journal of Symbolic Logic (JSL)  
We prove that a wide class 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.  ... 
doi:10.1017/jsl.2015.46 fatcat:jmb7i4u5rnfp3luuavikvadg2q

Indestructible Guessing Models and the Continuum [article]

Sean Cox, John Krueger
2019 arXiv   pre-print
We introduce a stronger version 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 .  ... 
arXiv:1510.05297v3 fatcat:f34xjkqcrfahdotmh5wc4tl7pi

Coding with ladders a well-ordering of the reals [article]

Uri Abraham, Saharon Shelah
2001 arXiv   pre-print
Any 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.  ... 
arXiv:math/0104195v1 fatcat:yeadxlhzhzhohe6njqatwxhuq4

Coding with ladders a well ordering of the reals

Uri Abraham, Saharon Shelah
2002 Journal of Symbolic Logic (JSL)  
change at will the 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.  ... 
doi:10.2178/jsl/1190150099 fatcat:plneaw7xlnas7gsucwikcd3nve

JSL volume 81 issue 1 Cover and Front matter

2016 Journal of Symbolic Logic (JSL)  
The BULLETIN, the JOURNAL 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  ... 
doi:10.1017/jsl.2016.2 fatcat:57zgexyc7fhshcs7f6ia4fz6w4

On the consistency strength of the proper forcing axiom

Matteo Viale, Christoph Weiß
2011 Advances in Mathematics  
Using this, we argue to show that any 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.  ... 
doi:10.1016/j.aim.2011.07.016 fatcat:jb4iwnxqevcazbeo7wzkvufmym

Guessing models and the approachability ideal [article]

Rahman Mohammadpour, Boban Velickovic
2019 arXiv   pre-print
This principle implies ISP(ω_2) 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 ).  ... 
arXiv:1802.10125v2 fatcat:c3nduoznjzcypc4fud5pvdzluu

Forcing axioms, approachability, and stationary set reflection [article]

Sean D. Cox
2020 arXiv   pre-print
We also adapt some ideas 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.  ... 
arXiv:1807.06129v3 fatcat:wo7ath4exfddbc6wswfght3axu

Rado's conjecture and its Baire version [article]

Jing Zhang
2019 arXiv   pre-print
In particular, we show a fragment 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.  ... 
arXiv:1712.02455v4 fatcat:j64m432cu5egjfflmqk4dn6a7a

Joint Laver diamonds and grounded forcing axioms [article]

Miha E. Habič
2017 arXiv   pre-print
I explore two separate topics: the concept 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.  ... 
arXiv:1705.04422v1 fatcat:cw7z423bcbby5ezi67jnz2mo7i

SPM Bulletin 20 [article]

Boaz Tsaban
2007 arXiv   pre-print
; covering by translates 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.  ... 
arXiv:math/0703894v1 fatcat:tynedt26pfcrjb7n2drjatvumu

Implications for climate sensitivity from the response to individual forcings

Kate Marvel, Gavin A. Schmidt, Ron L. Miller, Larissa S. Nazarenko
2015 Nature Climate Change  
Climate 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  ... 
doi:10.1038/nclimate2888 fatcat:wlv3x6rq55gdto5u777trtz47i

Isogeny in Superstable Groups [article]

James Freitag
2011 arXiv   pre-print
We study 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  ... 
arXiv:1110.1766v1 fatcat:3625cvzapjgj3al4gmc5lg53v4

Definable well-orders of H(ω 2) and GCH

David Asperó, Sy-David Friedman
2012 Journal of Symbolic Logic (JSL)  
Assuming 2ℵ0 = ℵ1 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.  ... 
doi:10.2178/jsl.7704030 fatcat:hglrqhdywjb3hm2czqa5jmiqmq
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