Mirna Dzamonja: On wide Aronszajn trees
04/11/20 18:48
Tuesday, November 10, 2020 17:00
Location: Zoom.us: if you want to participate please contact organizers
Speaker: Mirna Dzamonja (CNRS & Pantheon-Sorbonne University & Czech Academy of Sciences)
Title: On wide Aronszajn trees
Abstract: Aronszajn trees are a staple of set theory, but there are applications where the requirement of all levels being countable is of no importance. This is the case in set-theoretic model theory, where trees of height and size \(\omega_1\) but with no uncountable branches play an important role by being clocks of Ehrenfeucht--Fraïssé games that measure similarity of model of size \(\aleph_1\). We call such trees wide Aronszajn. In this context one can also compare trees T and T’ by saying that T weakly embeds into T’ if there is a function f that map T into T’ while preserving the strict order \(<_T\). This order translates into the comparison of winning strategies for the isomorphism player, where any winning strategy for T’ translates into a winning strategy for T’. Hence it is natural to ask if there is a largest such tree, or as we would say, a universal tree for the class of wide Aronszajn trees with weak embeddings. It was known that there is no such a tree under CH, but in 1994 Mekler and Väänanen conjectured that there would 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 universal wide Aronszajn tree.
The talk will discuss that paper. The paper is available on the arxiv and on line at JSL in the preproof version DOI: 10.1017/jsl.2020.42
Location: Zoom.us: if you want to participate please contact organizers
Speaker: Mirna Dzamonja (CNRS & Pantheon-Sorbonne University & Czech Academy of Sciences)
Title: On wide Aronszajn trees
Abstract: Aronszajn trees are a staple of set theory, but there are applications where the requirement of all levels being countable is of no importance. This is the case in set-theoretic model theory, where trees of height and size \(\omega_1\) but with no uncountable branches play an important role by being clocks of Ehrenfeucht--Fraïssé games that measure similarity of model of size \(\aleph_1\). We call such trees wide Aronszajn. In this context one can also compare trees T and T’ by saying that T weakly embeds into T’ if there is a function f that map T into T’ while preserving the strict order \(<_T\). This order translates into the comparison of winning strategies for the isomorphism player, where any winning strategy for T’ translates into a winning strategy for T’. Hence it is natural to ask if there is a largest such tree, or as we would say, a universal tree for the class of wide Aronszajn trees with weak embeddings. It was known that there is no such a tree under CH, but in 1994 Mekler and Väänanen conjectured that there would 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 universal wide Aronszajn tree.
The talk will discuss that paper. The paper is available on the arxiv and on line at JSL in the preproof version DOI: 10.1017/jsl.2020.42