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The borel conjecture

Haim Judah, Saharon Shelah, W.H. Woodin
1990 Annals of Pure and Applied Logic  
We show the Bore1 Conjecture is consistent with the continuum large. Introduction In this work we consider the problem of the Bore1 Conjecture with large continuum.  ...  The consistency of the Bore1 Conjecture was proved by Laver [8], from the consistency of Zermelo-Fraenkel set theory.  ...  We will finish with the following conjecture: Conjecture. V k "Bore1 Conjecture" iff VR k "Bore1 Conjecture" when R is random forcing. 0. 2 . 2 Definition.  ... 
doi:10.1016/0168-0072(90)90058-a fatcat:gsm4dmujtfhxpm5har36imlhj4

The γ-borel conjecture

Arnold W. Miller
2004 Archive for Mathematical Logic  
Paul Szeptycki asked if it was possible to have a sort of weak Borel conjecture be true, i.e., every γ-set countable, while the Borel conjecture is false. We answer his question positively.  ...  Laver [8] showed that it is relatively consistent with ZFC that the Borel conjecture is true, i.e., every strong measure zero set is countable.  ...  C -BC to be the statement that every set of reals with the property C is countable and let SMZ-BC denote the standard Borel conjecture, every strong measure zero set is countable.  ... 
doi:10.1007/s00153-004-0260-0 fatcat:uld2r4su7nbxnbpimzv5a7s5re

The gamma - Borel conjecture [article]

Arnold W. Miller
2003 arXiv   pre-print
On the other hand every strong measure zero set is countable iff every set with the Rothberger property is countable.  ...  An open cover is a gamma-cover iff every element of the space is in all but finitely many elements of the cover. Gerlits and Nagy proved that every gamma-set has strong measure zero.  ...  C ′′ -BC to be the statement that every set of reals with the property C ′′ is countable and let SMZ-BC denote the standard Borel conjecture, every strong measure zero set is countable.  ... 
arXiv:math/0312308v1 fatcat:dka3n6ai5ndwdd75nfrzuwyog4

An overview of the proof in "Borel Conjecture and Dual Borel Conjecture" [article]

Martin Goldstern and Jakob Kellner and Saharon Shelah and Wolfgang Wohofsky
2011 arXiv   pre-print
This note gives an informal overview of the proof in our paper "Borel Conjecture and Dual Borel Conjecture", see arXiv:1105.0823.  ...  X + F is non-null in the extension by R * P ω 2 (where the Borel code F is evaluated in the extension).  ...  X + F is forced to be a subset of a null set (or rather, a Borel code) Z; this already has to happen 7 at some stage β < ω 2 .  ... 
arXiv:1112.4424v1 fatcat:azwiodcm3vgs5nmvq4djt2oxgm

Borel conjecture and dual Borel conjecture

Martin Goldstern, Jakob Kellner, Saharon Shelah, Wolfgang Wohofsky
2013 Transactions of the American Mathematical Society  
We show that it is consistent that the Borel Conjecture and the dual Borel Conjecture hold simultaneously.  ...  revision:2011-12-27 modified:2011-12-28 BOREL CONJECTURE AND DUAL BOREL CONJECTURE or: "nice" in the sense of [Kun80, 5.11]  ...  AND DUAL BOREL CONJECTURE revision:2011-12-27 modified:2011-12-28 BOREL CONJECTURE AND DUAL BOREL CONJECTURE constructed inLemma 3.18 I.e., j β (x, p↾β) = j α (x, p↾β) = j α (x, p)↾β  ... 
doi:10.1090/s0002-9947-2013-05783-2 fatcat:qgmwnoc2obf6tl3xeqlltn6lde

Ramsey Theory and the Borel Conjecture [article]

Marion Scheepers
2019 arXiv   pre-print
The current state of knowledge about the connection between the Borel covering property and Ramsey theory is outlined in this paper.  ...  Initially the connection is established for the situation when the set with the Borel covering property is a proper subset of a σ-compact uniform space.  ...  principle S 1 (O, O) , then X has the Borel covering property.  ... 
arXiv:1912.03796v1 fatcat:ybw2nit5vfd2hggu2xhldy7lda

The Borel/Novikov conjectures and stable diffeomorphisms of 4-manifolds [article]

James F. Davis
2011 arXiv   pre-print
This injectivity is implied by the Borel/Novikov conjecture for torsion-free groups, which is known for many groups.  ...  in Z/2 to the L-theory of the group is injective.  ...  the Borel/Novikov conjectures and four-dimensional topology.  ... 
arXiv:math/0406084v4 fatcat:itndu7zv3bfzbiyuzkfvlqaxka

The Generalized Borel Conjecture and Strongly Proper Orders

Paul Corazza
1989 Transactions of the American Mathematical Society  
The Generalized Borel Conjecture is the statement that C = [R] <c . We show that this statement is also independent.  ...  The Borel Conjecture is the statement that C = [R] <ω 1 , where C is the class of strong measure zero sets; it is known to be independent of ZFC.  ...  The Borel Conjecture is the statement C = [R] <ω1 .  ... 
doi:10.2307/2001276 fatcat:jhzcb47mbnbi7jbesl5jegltxq

The generalized Borel conjecture and strongly proper orders

Paul Corazza
1989 Transactions of the American Mathematical Society  
The Generalized Borel Conjecture is the statement that C = [R] <c . We show that this statement is also independent.  ...  The Borel Conjecture is the statement that C = [R] <ω 1 , where C is the class of strong measure zero sets; it is known to be independent of ZFC.  ...  The Borel Conjecture is the statement C = [R] <ω1 .  ... 
doi:10.1090/s0002-9947-1989-0982239-x fatcat:fpy3g34hgfd5zbccs4pcwrrhpa

The Borel Conjecture for hyperbolic and CAT(0)-groups [article]

Arthur Bartels, Wolfgang Lueck
2010 arXiv   pre-print
We prove the Borel Conjecture for a class of groups containing word-hyperbolic groups and groups acting properly, isometrically and cocompactly on a finite dimensional CAT(0)-space.  ...  We are indebted to Tom Farrell and Lowell Jones for their wonderful conjecture and work surrounding it.  ...  The work was financially supported by the Sonderforschungsbereich 478 -Geometrische Strukturen in der Mathematik -and the Max-Planck-Forschungspreis and the Leibniz-Preis of the second author.  ... 
arXiv:0901.0442v2 fatcat:hsns2v44njb2ddqc5lrk7avj6e

The Borel Conjecture for hyperbolic and CAT(0)-groups

Arthur Bartels, Wolfgang Lück
2012 Annals of Mathematics  
We prove the Borel Conjecture for a class of groups containing wordhyperbolic groups and groups acting properly, isometrically and cocompactly on a finite-dimensional CAT(0)-space.  ...  These conjectures are the key to the Borel Conjecture. See [42] for a survey on the Farrell-Jones Conjectures.  ...  The spheres S n are topologically rigid as predicted by the Poincaré Conjecture. We will focus on the Borel Conjecture which asserts: Closed aspherical manifolds are topologically rigid.  ... 
doi:10.4007/annals.2012.175.2.5 fatcat:psljre5bjrckvm2w5ko4cxl3qy

On the Mori-Szekely conjectures for the Borel-Cantelli lemma [article]

Chunrong Feng, Liangpan Li
2013 arXiv   pre-print
The purpose of this note is to show by constructing counterexamples that two conjectures of Móri and Székely for the Borel-Cantelli lemma are false.  ...  The purpose of this note is to show that both conjectures are false.  ...  Borel-Cantelli lemma, Gallot-Kounias bound, Kuai-Alajaji-Takahara bound.  ... 
arXiv:1304.0489v1 fatcat:q5j2mxu2rjb5foc53gcmisbid4

Three-dimensional surgery theory, UNil-groups and the Borel conjecture

Bjørn Jahren, Sławomir Kwasik
2003 Topology  
Applications leading to new results and conjectures concerning the Borel conjecture in dimension ¿ 5 and UNil-groups are also discussed. ?  ...  We present almost complete computations of the surgery obstruction for 3-manifolds, as well as consequences for a suitable version of the structure set in dimension three.  ...  Acknowledgements The authors would like to thank the referee for pointing out a misstatement in the proof of Theorem 15(b) in the ÿrst version of the paper.  ... 
doi:10.1016/s0040-9383(03)00003-x fatcat:ti5pb7vtmzckbenilsrl4vmrwa

There are no $Q$-points in Laver's model for the Borel conjecture

Arnold W. Miller
1980 Proceedings of the American Mathematical Society  
In Laver's model N for the Borel conjecture [L] there are no semi-Q-points. Proof.  ...  Conjecture. Borel conjecture <=> there does not exist a semi-g-point.  ... 
doi:10.1090/s0002-9939-1980-0548093-2 fatcat:bmaxk6enu5cxhgszmtovaquo6u

There are no Q-Points in Laver's Model for the Borel Conjecture

Arnold W. Miller
1980 Proceedings of the American Mathematical Society  
PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 78, Number 1, January 1980 THERE ARE NO Q-POINTS IN LAVER'S MODEL FOR THE BOREL CONJECTURE ARNOLD W. MILLER ABSTRACT.  ...  CONJECTURE. Borel conjecture < there does not exist a semi-Q-point. THEOREM 5 . 5 x V = {A c o X : {n: {m: (n, m) E A) E V} C U).  ... 
doi:10.2307/2043048 fatcat:yk6tobqko5eo3etnwbrs34hxki
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