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# Documents  03E05 | enregistrements trouvés : 37

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## Distributive Aronszajn trees Rinot, Assaf | CIRM H

Post-edited

Research talks;Logic and Foundations

It is well-known that the statement "all $\aleph_1$-Aronszajn trees are special'' is consistent with ZFC (Baumgartner, Malitz, and Reinhardt), and even with ZFC+GCH (Jensen). In contrast, Ben-David and Shelah proved that, assuming GCH, for every singular cardinal $\lambda$: if there exists a $\lambda^+$-Aronszajn tree, then there exists a non-special one. Furthermore:
Theorem (Ben-David and Shelah, 1986) Assume GCH and that $\lambda$ is singular cardinal. If there exists a special $\lambda^+$-Aronszajn tree, then there exists a $\lambda$-distributive $\lambda^+$-Aronszajn tree.
This suggests that following stronger statement:
Conjecture. Assume GCH and that $\lambda$ is singular cardinal.
If there exists a $\lambda^+$-Aronszajn tree,
then there exists a $\lambda$-distributive $\lambda^+$-Aronszajn tree.

The assumption that there exists a $\lambda^+$-Aronszajn tree is a very mild square-like hypothesis (that is, $\square(\lambda^+,\lambda)$).
In order to bloom a $\lambda$-distributive tree from it, there is a need for a toolbox, each tool taking an abstract square-like sequence and producing a sequence which is slightly better than the original one.
For this, we introduce the monoid of postprocessing functions and study how it acts on the class of abstract square sequences.
We establish that, assuming GCH, the monoid contains some very powerful functions. We also prove that the monoid is closed under various mixing operations.
This allows us to prove a theorem which is just one step away from verifying the conjecture:

Theorem 1. Assume GCH and that $\lambda$ is a singular cardinal.
If $\square(\lambda^+,<\lambda)$ holds, then there exists a $\lambda$-distributive $\lambda^+$-Aronszajn tree.
Another proof, involving a 5-steps chain of applications of postprocessing functions, is of the following theorem.

Theorem 2. Assume GCH. If $\lambda$ is a singular cardinal and $\square(\lambda^+)$ holds, then there exists a $\lambda^+$-Souslin tree which is coherent mod finite.

This is joint work with Ari Brodsky. See: http://assafrinot.com/paper/29
It is well-known that the statement "all $\aleph_1$-Aronszajn trees are special'' is consistent with ZFC (Baumgartner, Malitz, and Reinhardt), and even with ZFC+GCH (Jensen). In contrast, Ben-David and Shelah proved that, assuming GCH, for every singular cardinal $\lambda$: if there exists a $\lambda^+$-Aronszajn tree, then there exists a non-special one. Furthermore:
Theorem (Ben-David and Shelah, 1986) Assume GCH and that $\lambda$ is singular ...

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## Chain conditions, unbounded colorings and the $C$-sequence spectrum Rinot, Assaf | CIRM H

Post-edited

Research talks

The productivity of the $\kappa$-chain condition, where $\kappa$ is a regular, uncountable cardinal, has been the focus of a great deal of set-theoretic research. In the 1970’s, consistent examples of $kappa-cc$ posets whose squares are not $\kappa-cc$ were constructed by Laver, Galvin, Roitman and Fleissner. Later, ZFC examples were constructed by Todorcevic, Shelah, and others. The most difficult case, that in which $\kappa = \aleph{_2}$, was resolved by Shelah in 1997.
In the first part of this talk, we shall present analogous results regarding the infinite productivity of chain conditions stronger than $\kappa-cc$. In particular, for any successor cardinal $\kappa$, we produce a ZFC example of a poset with precaliber $\kappa$ whose $\omega ^{th}$ power is not $\kappa-cc$. To do so, we introduce and study the principle $U(\kappa , \mu , \theta , \chi )$ asserting the existence of a coloring $c:\left [ \kappa \right ]^{2}\rightarrow \theta$ satisfying a strong unboundedness condition.
In the second part of this talk, we shall introduce and study a new cardinal invariant $\chi \left ( \kappa \right )$ for a regular uncountable cardinal $\kappa$ . For inaccessible $\kappa$, $\chi \left ( \kappa \right )$ may be seen as a measure of how far away $\kappa$ is from being weakly compact. We shall prove that if $\chi \left ( \kappa \right )> 1$, then $\chi \left ( \kappa \right )=max(Cspec(\kappa ))$, where:
(1) Cspec$(\kappa)$ := {$\chi (\vec{C})\mid \vec{C}$ is a sequence over $\kappa$} $\setminus \omega$, and
(2) $\chi \left ( \vec{C} \right )$ is the least cardinal $\chi \leq \kappa$ such that there exist $\Delta\in\left [ \kappa \right ]^{\kappa }$ and
b : $\kappa \rightarrow \left [ \kappa \right ]^{\chi }$ with $\Delta \cap \alpha \subseteq \cup _{\beta \in b(\alpha )}C_{\beta }$ for every $\alpha < \kappa$.
We shall also prove that if $\chi (\kappa )=1$, then $\kappa$ is greatly Mahlo, prove the consistency (modulo the existence of a supercompact) of $\chi (\aleph_{\omega +1})=\aleph_{0}$, and carry a systematic study of the effect of square principles on the $C$-sequence spectrum.
In the last part of this talk, we shall unveil an unexpected connection between the two principles discussed in the previous parts, proving that, for infinite regular cardinals $\theta< \kappa ,\theta \in Cspec(\kappa )$ if there is a closed witness to $U_{(\kappa ,\kappa ,\theta ,\theta )}$.
This is joint work with Chris Lambie-Hanson.
The productivity of the $\kappa$-chain condition, where $\kappa$ is a regular, uncountable cardinal, has been the focus of a great deal of set-theoretic research. In the 1970’s, consistent examples of $kappa-cc$ posets whose squares are not $\kappa-cc$ were constructed by Laver, Galvin, Roitman and Fleissner. Later, ZFC examples were constructed by Todorcevic, Shelah, and others. The most difficult case, that in which $\kappa = \aleph{_2}$, ...

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## Graph theory, singapore 1983 :proceedings of the first southeast asian graph theory colloquium,held in singapore#May 10-28 Koh, K. M. ; Yap, H. P. | Springer-Verlag 1984

Congrès

- 325 p.
ISBN 978-0-387-13368-3

Lecture notes in mathematics , 1073

Localisation : Collection 1er étage

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## Axiomatic set theory :proceedings of the ams-ims-siam joint summer research conference in the mathematics sciences on... held at the university of colorado#June 19-25 Baumgartner, James E. ; Martin, Donald A. ; Shelah, Saharon | American Mathematical Society 1984

Congrès

- 259 p.
ISBN 978-0-8218-5026-8

Contemporary mathematics , 0031

Localisation : Collection 1er étage

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## Abelian group theory and related topics :conference on abelian groups#Aug. 1-7 Gobel, Rudiger ; Hill, Paul ; Liebert, Wolfgang | American Mathematical Society 1994

Congrès

- 432 p.
ISBN 978-0-8218-5178-4

Contemporary mathematics , 0171

Localisation : Collection 1er étage

GCH # K indice O d'anneau régulier # Richard Scott Pierce # algèbre de Artin héréditaire # anneau de valuation discret incomplet # anneau des nombres universels # anneau minimal # contraction # critère de scission # décomposition # endomorphisme et automorphisme # ensemble de cotypes # ensemble de types # foncteur # gerbe de Pierce # groupe B indice 2 # groupe abélien p puissance alpha injectif # groupe abélien presque complètement décomposable # groupe de Baer-Specker # groupe de Butler de rang infini # groupe de Warfield local # groupe de Whitehead non libre # groupe de nombres # groupe libre de torsion # groupe multiplicatif d'un corps # idempotent central # index borné de nilpotence # module noethérien # nombre premier # part divisible de groupe quotient # présentation finie des modules # radical de Jacobson # représentation et dualité # sommande direct # sous-groupe pur # théorie de la mesure # théorie des groupes abéliens # torsion dans les quotients # union de chaînes GCH # K indice O d'anneau régulier # Richard Scott Pierce # algèbre de Artin héréditaire # anneau de valuation discret incomplet # anneau des nombres universels # anneau minimal # contraction # critère de scission # décomposition # endomorphisme et automorphisme # ensemble de cotypes # ensemble de types # foncteur # gerbe de Pierce # groupe B indice 2 # groupe abélien p puissance alpha injectif # groupe abélien presque complètement décomposable ...

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## Advances in logic :the north texas logic conference#Oct. 8-10 Gao, Su ; Jackson, Steve ; Zhang, Yi | Amercian Mathematical Society 2007

Congrès

- 150 p.
ISBN 978-0-8218-3819-8

Contemporary mathematics , 0425

Localisation : Collection 1er étage

logique

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## Ultrafilters across mathematics.International congress ULTRAMATH 2008: applications of ultrafilters and ultraproducts in mathematicsPisa # june 1-7, 2008 Bergelson, Vitaly ; Blass, Andreas ; Di Nasso, Mauro ; Jin, Renling | American Mathematical Society 2010

Congrès

- ix; 200 p.
ISBN 978-0-8218-4833-3

Contemporary mathematics , 0530

Localisation : Collection 1er étage

logique mathématique # ensemble combinatoire # théorie de Ramsey # ultrafiltres

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## Reflection of clubs, and forcing principles at $\aleph_2$ Neeman, Itay | CIRM

Multi angle

Research talks;Logic and Foundations

In 1971 Baumgartner showed it is consistent that any two $\aleph_1$-dense subsets of the real line are order isomorphic. This was important both for the methods of the proof and for consequences of the result. We introduce methods that lead to an analogous result for $\aleph_2$-dense sets.

Keywords : forcing - large cardinals - Baumgartner isomorphism - infinitary Ramsey principles - reflection principles

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## Reflection of stationary sets and the tree property at $\aleph_{\omega^2+1}$ Fontanella, Laura | CIRM

Multi angle

Research talks;Logic and Foundations

The combinatorics of successors of singular cardinals presents a number of interesting open problems. We discuss the interactions at successors of singular cardinals of two strong combinatorial properties, the stationary set reflection and the tree property. Assuming the consistency of infinitely many supercompact cardinals, we force a model in which both the stationary set reflection and the tree property hold at $\aleph_{\omega^2+1}$. Moreover, we prove that the two principles are independent at this cardinal, indeed assuming the consistency of infinitely many supercompact cardinals it is possible to force a model in which the stationary set reflection holds, but the tree property fails at $\aleph_{\omega^2+1}$. This is a joint work with Menachem Magidor.
Keywords : forcing - large cardinals - successors of singular cardinals - stationary reflection - tree property
The combinatorics of successors of singular cardinals presents a number of interesting open problems. We discuss the interactions at successors of singular cardinals of two strong combinatorial properties, the stationary set reflection and the tree property. Assuming the consistency of infinitely many supercompact cardinals, we force a model in which both the stationary set reflection and the tree property hold at $\aleph_{\omega^2+1}$. ...

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## Some results on set mappings Komjáth, Péter | CIRM H

Multi angle

Research talks

I give a survey of some recent results on set mappings.

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## Lower bounds for the perfect set property at weakly compact cardinals Müller, Sandra | CIRM H

Multi angle

Research talks

By the Cantor-Bendixson theorem, subtrees of the binary tree on $\omega$ satisfy a dichotomy - either the tree has countably many branches or there is a perfect subtree (and in particular, the tree has continuum manybranches, regardless of the size of the continuum). We generalize this to arbitrary regular cardinals $\kappa$ and ask whether every $\kappa$-tree with more than $\kappa$ branches has a perfect subset. From large cardinals, this statement isconsistent at a weakly compact cardinal $\kappa$. We show using stacking mice that the existence of a non-domestic mouse (which yields a model with a proper class of Woodin cardinals and strong cardinals) is a lower bound. Moreover, we study variants of this statement involving sealed trees, i.e. trees with the property that their set of branches cannot be changed by certain forcings, and obtain lower bounds for these as well. This is joint work with Yair Hayut. By the Cantor-Bendixson theorem, subtrees of the binary tree on $\omega$ satisfy a dichotomy - either the tree has countably many branches or there is a perfect subtree (and in particular, the tree has continuum manybranches, regardless of the size of the continuum). We generalize this to arbitrary regular cardinals $\kappa$ and ask whether every $\kappa$-tree with more than $\kappa$ branches has a perfect subset. From large cardinals, this ...

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## Principia mathematica. Vol.II Whitehead, Alfred North ; Russell, Bertrand | Cambridge University Press 1927

Ouvrage

- 742 p.
ISBN 978-0-521-06791-1

Localisation : Ouvrage RdC (WHIT)

arithmétique cardinale # convergence # limite de fonction # principe de mathématiques # produit de relations # relation # théorie des séries # Whitehead # oeuvres complètes

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## Combinatorial set theory Williams, Neil H. | North-Holland Publishing Co. 1977

Ouvrage

- 208 p.
ISBN 978-0-7204-0722-8

Studies in logic and the foundations of mathematics , 0091

Localisation : Ouvrage RdC (WILL)

familles presque disjointes d'ensembles # graphe infini # nombre cardinal # propriété de décomposition ou d'intersection des familles d' # relation de partition ordinaire # relation de partition ordinaire pour nombre ordinal # relation de partition polarisée # théorie combinatoire des ensembles # transformation d'ensembles

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## Proper forcing Shelah, Saharon | Springer-Verlag 1982

Ouvrage

ISBN 978-3-540-11593-9

Lecture notes in mathematics , 0940

Localisation : Collection 1er étage

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## Combinatoricsset systems, hypergraphs, families of vectors and combinatorial probability Bollobas, Bela | Cambridge University Press 1985

Ouvrage

- 431 p.
ISBN 978-0-521-33703-8

Localisation : Ouvrage RdC (BOLL)

combinatoire # hypergraphe # semble

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## Injective choice functions Holz, Michael ; Podewski, Klaus-Peter ; Steffens , Karsten | Springer-Verlag 1987

Ouvrage

- 183 p.
ISBN 978-3-540-17221-5

Lecture notes in mathematics , 1238

Localisation : Collection 1er étage

analyse combinatoire # combinatoire # famille # problème de combinatoire # semble # structure # théorie des sembles # théorie des sembles combinatoires

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## Topics in set theory Bekkali, M. | Springer-Verlag 1991

Ouvrage

- 120 p.
ISBN 978-3-540-54121-9

Lecture notes in mathematics , 1476

Localisation : Collection 1er étage

axionie # cardinalité # cardinaux # mesurabilité de Lebesque # théorie des ensembles combinatoires # théorie des ensembles

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## Partition problems intopology Todorcevic, Stevo | American Mathematical Society 1989

Ouvrage

- 116 p.
ISBN 978-0-8218-5091-6

Contemporary mathematics , 0084

Localisation : Réserve

théorie des ensembles # topologie

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## On sets not belonging to algebras of subsets Grinblat, L. S. | American Mathematical Society 1993

Ouvrage

ISBN 978-0-8218-2541-9

Memoirs of the american mathematical society , 0480

Localisation : Collection 1er étage

algèbre de sousensemble # compactification de Stone-Cech # mesure à 2 valeurs # théorie combinatoire des ensembles # ultra filtre

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## Partition problems in topology Todorcevic, Stevo | American Mathematical Society 1989

Ouvrage

- 116 p.
ISBN 978-0-8218-5091-6

Contemporary mathematics , 0084

Localisation : Collection 1er étage

combinatoire # problème de Ramsey # théorie des ensembles # topologie

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