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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 ...

03E05 ; 03E65 ; 03E35 ; 05C05

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- 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

03E15 ; 03E17 ; 03E25 ; 03E35 ; 03E50 ; 03E60 ; 03E65 ; 26A21 ; 28A05 ; 28A99 ; 54D99 ; 54G15 ; 54H05 ; 26-02 ; 26A03 ; 03E75

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- xiv; 502 p.
ISBN 978-0-8218-4813-5

Mathematical surveys and monographs , 0155

Localisation : Collection 1er étage

théorie descriptive des ensembles # axiome # fonction récursive # hierarchie # classes d'ensembles # ensemble de Borel # ensemble projectif

03-02 ; 03D55 ; 03E15 ; 28A05 ; 54H05 ; 03E60 ; 03E65 ; 03E45 ; 03D20 ; 03D75 ; 26A21

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- 194 p.
ISBN 978-3-540-30989-5

Lecture notes in mathematics , 1876

Localisation : Collection 1er étage

axiome du choix

03E25 ; 03E60 ; 03E65 ; 05C15 ; 06B10 ; 08B30 ; 18A40 ; 26A03 ; 28A20 ; 46A22 ; 54Bxx ; 54C35 ; 54Dxx ; 91a35

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ISBN 978-0-8218-2117-6

Memoirs of the american mathematical society , 0702

Localisation : Collection 1er étage

algébre booléen # quotient analytique # théorie d'ensemble

03E15 ; 03E50 ; 03E65 ; 06E05 ; 06G05 ; 28A05 ; 54C05 ; 54D40

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