English
Distributive Aronszajn trees
Post-edited
Auteurs : Rinot, Assaf (Auteur de la Conférence)
CIRM (Editeur )
03E05 - Combinatorial set theory (logic)
03E35 - Consistency and independence results
03E65 - Other hypotheses and axioms (set theory)
05C05 - Trees
CIRM (Editeur )
Loading the player...
|
| square principle alternative constructions improving squares postprocessing functions mixing lemma |
Résumé :
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
Keywords : Aronszajn tree; uniformly coherent Souslin tree; walks on ordinals; club guessing; square principle; $C$-sequence; postprocessing function; distributive tree; fat set; nonspecial Aronszajn tree
Codes MSC : 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
Keywords : Aronszajn tree; uniformly coherent Souslin tree; walks on ordinals; club guessing; square principle; $C$-sequence; postprocessing function; distributive tree; fat set; nonspecial Aronszajn tree
03E05 - Combinatorial set theory (logic)
03E35 - Consistency and independence results
03E65 - Other hypotheses and axioms (set theory)
05C05 - Trees
|
Informations sur la Vidéo
Réalisateur : Hennenfent, GuillaumeLangue : Anglais Date de publication : 12/10/2017 Date de captation : 10/10/2017 Sous collection : Research talks arXiv category : Logic Domaine : Logic and Foundations Format : MP4 (.mp4) - HD Durée : 00:45:58 Audience : Researchers Download : https://videos.cirm-math.fr/2017-10-10_Rinot.mp4 
|
Informations sur la Rencontre
Nom de la rencontre : 14th International workshop in set theory / XIVe Atelier international de théorie des ensemblesOrganisateurs de la rencontre : Dzamonja, Mirna ; Magidor, Menachem ; Velickovic, Boban ; Woodin, W. Hugh Dates : 09/10/2017 - 13/10/2017 Année de la rencontre : 2017 URL Congrès : http://conferences.cirm-math.fr/1606.html
Données de citation
DOI : 10.24350/CIRM.V.19228603Citer cette vidéo: Rinot, Assaf (2017). Distributive Aronszajn trees. CIRM. Audiovisual resource. doi:10.24350/CIRM.V.19228603 URI : http://dx.doi.org/10.24350/CIRM.V.19228603 |
Voir aussi
- [Multi angle] The hierarchy of second-order set theories between GBC and KM and beyond / Auteur de la Conférence Hamkins, Joel David.
- [Multi angle] From forcing models to realizability models / Auteur de la Conférence Fontanella, Laura.
- [Multi angle] Monochromatic sumsets for colourings of $\mathbb{R}$ / Auteur de la Conférence Soukup, Daniel T..
- [Multi angle] Prikry type forcing and combinatorial properties / Auteur de la Conférence Sinapova, Dima.
Bibliographie
- Brodsky, A.M., & Rinot, A. (2017). Distributive Aronszajn trees - http://assafrinot.com/paper/29
Imagette Video
