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SUMMARY:Mirna Dzamonja (Université Paris 1)
DTSTART:20201021T140000Z
DTEND:20201021T153000Z
DTSTAMP:20260423T022035Z
UID:BCNSETS/4
DESCRIPTION:Title: <a href="https://researchseminars.org/talk/BCNSETS/4/">
 On Wide Aronszajn Trees</a>\nby Mirna Dzamonja (Université Paris 1) as pa
 rt of Barcelona Set Theory Seminar\n\n\nAbstract\nAronszajn trees are a st
 aple of set theory\, but there are applications where the requirement of a
 ll levels being countable is of no importance. This is the case in set-the
 oretic model theory\, where trees of height and size \\omega_1. but with n
 o uncountable branches play an important role by being clocks of Ehrenfeuc
 ht--Fraïssé games that measure similarity of model of size \\aleph_1. We
  call such trees wide Aronszajn. In this context one can also compare tree
 s T and T’ by saying that T weakly embeds into T’ if there is a functi
 on f that map T into T’ while preserving the strict order <_T. This orde
 r translates into the comparison of winning strategies for the isomorphism
  player\, where any winning strategy for T’ translates into a winning st
 rategy for T’. Hence it is natural to ask if there is a largest such tre
 e\, or as we would say\, a universal tree for the class of wood Aronszajn 
 trees with weak embeddings. It was known that there is no such a tree unde
 r CH\, but in 1994 Mekler and V\\"a\\"an\\"anen conjectured that there wou
 ld be under MA(\\omega_1). In our upcoming JSL  paper with Saharon Shelah 
 we prove that this is not the case: under MA(\\omega_1) there is no univer
 sal wide Aronszajn tree. The talk will discuss that paper. The paper is av
 ailable on the arxiv and on line at JSL in the preproof version DOI: 10.10
 17/jsl.2020.42\n
LOCATION:https://researchseminars.org/talk/BCNSETS/4/
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