Mirna Dzamonja
Mirna Dzamonja: Reasonable structures of size \(\aleph_1\)
13/05/22 08:27
Tuesday, May 17, 2022 17:00
Location: room 605, Mathematical Institute, University of Wroclaw
Speaker: Mirna Dzamonja (Université deParis-Cité)
Title: Reasonable structures of size \(\aleph_1\)
Abstract: We are interested to develop a theory of structures of size \(\aleph_1\) which are ’tame’ in the sense that they in some sense or other preserve the nice properties that we are used to seeing on the countable structures.
We explain the aim of the programme and then discuss a joint work with Wiesław Kubiś on a specific way of constructing structures of size \(\aleph_1\) using finite approximations, namely by organising the approximations along a simplified morass. We demonstrate a connection with Fraïssé limits and show that the naturally obtained structure of size \(\aleph_1\) is homogeneous. We give some examples of interesting structures constructed, such as a homogeneous antimetric space of size \(\aleph_1\). Finally, we comment on the situation when one Cohen real is added.
Location: room 605, Mathematical Institute, University of Wroclaw
Speaker: Mirna Dzamonja (Université deParis-Cité)
Title: Reasonable structures of size \(\aleph_1\)
Abstract: We are interested to develop a theory of structures of size \(\aleph_1\) which are ’tame’ in the sense that they in some sense or other preserve the nice properties that we are used to seeing on the countable structures.
We explain the aim of the programme and then discuss a joint work with Wiesław Kubiś on a specific way of constructing structures of size \(\aleph_1\) using finite approximations, namely by organising the approximations along a simplified morass. We demonstrate a connection with Fraïssé limits and show that the naturally obtained structure of size \(\aleph_1\) is homogeneous. We give some examples of interesting structures constructed, such as a homogeneous antimetric space of size \(\aleph_1\). Finally, we comment on the situation when one Cohen real is added.
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