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# Documents  03E35 | enregistrements trouvés : 36

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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;Logic and Foundations

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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## Set theory and its applications.Annual Boise extravaganza in set theoryBoise # 1995-2010 Babinkostova, L. ; Caicedo, A. E. ; Geschke, S. ; Scheepers, M. | American Mathematical Society 2011

Congrès

- xi; 330 p.
ISBN 978-0-8218-4812-8

Contemporary mathematics , 0533

Localisation : Collection 1er étage

théorie des ensembles # théorie de Ramsey

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## Set theory :the conference held in honor of Andras Hajnal at DIMACS center at Rutgers University#Oct. 15-17 Thomas, Simon | American Mathematical Society 2002

Congrès

- 160 p.
ISBN 978-0-8218-2786-4

DIMACS series in discrete mathematics and theorerical computer science , 0058

Localisation : Collection 1er étage

théorie des nombres # relation de partition # théorie descriptive des ensembles # grand cardinal # convergence des resultats # indépendance des résultats

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## Séminaire N. Bourbabki. Vol. 1968-69 :exposés 353-357 | Secretariat N. Bourbaki 1969

Congrès

- pag. mult.

Localisation : Séminaire 1er étage

2-sous-groupe de Sylow de groupe simple # Stallings # consistance en théorie de la mesure # décomposition de groupe en produit libre # forme modulaire # représentation l- adique # régularité d'hypersurface minimale # sous-ensemble analytique # variété banachique

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## Nonstandard models of arithmetic and set theory :AMS special session on...#Jan. 15-16 Enayat, Ali ; Kossak, Roman | American Mathematical Society 2004

Congrès

- 167 p.
ISBN 978-0-8218-3535-7

Contemporary mathematics , 0361

Localisation : Collection 1er étage

théorie des modèles # théorie des ensembles non-standard # arithmétique # équivalence récursive # analyse non-archimédienne

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## Foundations of mathematics.Logic at Harvard : essays in honor of W. Hugh Woodin's 60th birthdayCambridge # March 27-29, 2015 Caicedo, Andrés Eduardo ; Cummings, James ; Koellner, Peter ; Larson, Paul B. | American Mathematical Society 2017

Congrès

- xx; 322 p.
ISBN 978-1-4704-2256-1

Contemporary mathematics , 0690

Localisation : Collection 1er étage

W. Hugh Woodin # théorie des ensembles # grand cardinal # espace de Banach # récursion # philosophie

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## Computability theory and its applications :current trends and open problems proceedings of a 1999 AMS-IMS-SIAM joint summer research conference on ... held at University of Colorado#June 13-17 Cholak, peter ; Lempp, Steffen ; Lerman, Manuel ; Shore, Richard A. | American Mathematical Society 2000

Congrès

- 320 p.
ISBN 978-0-8218-1922-7

Contemporary mathematics , 0257

Localisation : Collection 1er étage

application de la calculabilité # arithmétique # arithmétique d"ordre élevé # degré # degré de Turin # ensemble récursivement énumérable # fonction calculable # logique # modèle nonstandard # numération # récurrence # réductibilité # théorie de récurrence # théorie des modèles # théorie descriptive des ensembles

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

Multi angle

Research talks;Combinatorics;Logic and Foundations

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

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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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## 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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## Prikry type forcing and combinatorial properties Sinapova, Dima | CIRM H

Multi angle

Research talks;Logic and Foundations

We will analyze consequences of various types of Prikry forcing on combinatorial properties at singular cardinals and their successors, focusing on weak square and simultaneous stationary reflection. The motivation is how much compactness type properties can be obtained at successors of singulars, and especially the combinatorics at $\aleph_{\omega+1}$.

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## Monochromatic sumsets for colourings of $\mathbb{R}$ Soukup, Daniel T. | CIRM H

Multi angle

Research talks;Combinatorics;Logic and Foundations

N. Hindman, I. Leader and D. Strauss proved that if $2^{\aleph_0}<\aleph_\omega$ then there is a finite colouring of $\mathbb{R}$ so that no infinite sumset $X+X$ is monochromatic. Now, we prove a consistency result in the other direction: we show that consistently relative to a measurable cardinal for any $c:\mathbb{R}\to r$ with $r$ finite there is an infinite $X\subseteq \mathbb{R}$ so that $c\upharpoonright X+X$ is constant. The goal of this presentation is to discuss the motivation, ideas and difficulties involving this result, as well as the open problems around the topic. Joint work with P. Komjáth, I. Leader, P. Russell, S. Shelah and Z. Vidnyánszky. N. Hindman, I. Leader and D. Strauss proved that if $2^{\aleph_0}<\aleph_\omega$ then there is a finite colouring of $\mathbb{R}$ so that no infinite sumset $X+X$ is monochromatic. Now, we prove a consistency result in the other direction: we show that consistently relative to a measurable cardinal for any $c:\mathbb{R}\to r$ with $r$ finite there is an infinite $X\subseteq \mathbb{R}$ so that $c\upharpoonright X+X$ is constant. The goal of this ...

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

Multi angle

Research talks;Logic and Foundations

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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## Universal algebra Grätzer, George | Springer-Verlag 1979

Ouvrage

- 581 p.
ISBN 978-0-387-90355-2

Localisation : Ouvrage RdC (GRAT)

algèbre universelle # algèbre partielle # algèbre libre # logique équationnelle # algèbre de Boole # exercice d'algèbre

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## The structure of the real line Bukovsky, Lev | Birkhäuser 2011

Ouvrage

- xiv; 536 p.
ISBN 978-3-0348-0005-1

Monografie matematyczne , 0071

Localisation : Ouvrage RdC (BUKO)

fonction réelle # topologue élémentaire # application à la théorie des ensembles

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## The stationary tower :notes on a course Larson, Paul B. ; Woodin, W. Hugh | American Mathematical Society 2004

Ouvrage

- 132 p.
ISBN 978-0-8218-3604-0

University lecture series , 0032

Localisation : Collection 1er étage

théorie des ensembles # théorie descriptive des ensembles # tour stationnaire # grand cardinal # théorème de Woodin # forcing # plongement

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## The higher infinite :large cardinals in set theory from their beginnings Kanamori, Akihiro | Springer-Verlag 1997

Ouvrage

- 536 p.
ISBN 978-3-540-57071-4

Perspectives in mathematical logic

Localisation : Ouvrage RdC (KANA)

arbre et structure # compacité # constructibilité # détermination # forcing et ensemble de réels # grands cardinaux # hypothèse forte # inaccessibilité # indescriptibilité # infini supérieur # jeu infini # mesurabilité # plongement # propriété de partition # théorie des ensembles

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## The covering property axiom, CPA :a combinatorial core of the iterated perfect set model Ciesielski, Krzysztof ; Pawlikowski, Janusz | Cambridge University Press 2004

Ouvrage

- 174 p.
ISBN 978-0-521-83920-4

Cambridge tracts in mathematics , 0164

Localisation : Collection 1er étage;Réserve

CPA # axiome de recouvrement # axiome ZFC # modèle itéré de Sachs # jeu # prisme # modèle de Sacks

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## Set-theory:Boolean-valued models and independence proofs Bell, John l. | Clarendon Press;Oxford University Press 2005

Ouvrage

- xxii-191 p.
ISBN 978-0-19-960916-1

Oxford logic guides , 0047

Localisation : Ouvrage RdC (BELL)

logique mathématique # axiome de choix # algèbre de Heyting # algèbre booléenne # consistance # indépendance # hypothèse du continuum # axiome de Martin # problème de Souslin

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