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Distributive Aronszajn trees
Post-edited
Authors : Rinot, Assaf (Author of the conference)
CIRM (Publisher )
03E05 - Combinatorial set theory (logic)
03E35 - Consistency and independence results
03E65 - Other hypotheses and axioms (set theory)
05C05 - Trees
CIRM (Publisher )
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| square principle alternative constructions improving squares postprocessing functions mixing lemma |
Abstract :
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
MSC Codes : 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
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Information on the Video
Film maker : Hennenfent, GuillaumeLanguage : English Available date : 12/10/2017 Conference Date : 10/10/2017 Subseries : Research talks arXiv category : Logic Mathematical Area(s) : Logic and Foundations Format : MP4 (.mp4) - HD Video Time : 00:45:58 Targeted Audience : Researchers Download : https://videos.cirm-math.fr/2017-10-10_Rinot.mp4 
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Information on the Event
Event Title : 14th International workshop in set theory / XIVe Atelier international de théorie des ensemblesEvent Organizers : Dzamonja, Mirna ; Magidor, Menachem ; Velickovic, Boban ; Woodin, W. Hugh Dates : 09/10/2017 - 13/10/2017 Event Year : 2017 Event URL : http://conferences.cirm-math.fr/1606.html
Citation Data
DOI : 10.24350/CIRM.V.19228603Cite this video as: 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 |
See Also
- [Multi angle] The hierarchy of second-order set theories between GBC and KM and beyond / Author of the conference Hamkins, Joel David.
- [Multi angle] From forcing models to realizability models / Author of the conference Fontanella, Laura.
- [Multi angle] Monochromatic sumsets for colourings of $\mathbb{R}$ / Author of the conference Soukup, Daniel T..
- [Multi angle] Prikry type forcing and combinatorial properties / Author of the conference Sinapova, Dima.
Bibliography
- Brodsky, A.M., & Rinot, A. (2017). Distributive Aronszajn trees - http://assafrinot.com/paper/29
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