June 2020

# Sakae Fuchino: A/the (possible) solution of the Continuum Problem

18/06/20 10:43

Tuesday, June 23, 2020 17:15

The continuum (\(=2^{\aleph_0}\)) is either 1. \(\aleph_1\) or 2. \(\aleph_2\) or 3. fairly large.

Here, the fair largeness of the continuum can be expressed either in terms of weak mahloness and/or some other ``large'' cardinal notions compatible with the continuum, or even in terms of existence of some saturated ideals.

The reflection principles we consider here can be formulated as the following type of Downward Löwenheim-Skolem Theorems:

1'. For any structure A of countable signature, there is an elementary substructure B of A of cardinality \(<\aleph_2\) in terms of stationary logic.

2'. For any structure A of countable signature, there is an elementary substructure B of A of cardinality \(<2^{\aleph_0}\) in terms of stationary logic but only for formulas without free second order variables.

3'. For any structure A of countable signature, there is an elementary substructure B of A of cardinality \(<2^{\aleph_0}\) in terms of PKL logic (a variant of the stationary logic) in weak interpretation.

The reflection points \(<\aleph_2\) and \(<2^{\aleph_0}\) can be considered to be natural/necessary since the reflection down to \(<\aleph_2\) declares that \(\aleph_1\) strongly represents the situation of uncountability; the reflection down to \(<2^{\aleph_0}\) can be interpreted in the way that the reflection manifests that the continuum is very "rich".

The Downward Löwenheim-Skolem Theorems in terms of stationary logics can be also regarded as very natural principles: They can be characterized in terms of Diagonal Reflection Principles of Sean Cox.

Analyzing these three scenarios, we obtain the notion of Laver-generically large cardinals.

Existence of a Laver-generically supercompact cardinal

1''. for \(\sigma\)-closed pos implies 1'.;

2''. for proper pos implies 2'.; while the existence of a Laver-generically supercompact cardinal

3''. for ccc pos implies 3'.

The symmetry of the arguments involved suggests the possibility that the trichotomy might be a set-theoretic multiversal necessity.

If time allows, I shall also discuss about the reflection of non-metrizability of topological spaces, Rado's Conjecture and Galvin's Conjecture in connection with the reflection properties in 1., 2. and 3.

Most of the results to be presented here are obtained in a joint work with Hiroshi Sakai and André Ottenbreit Maschio Rodrigues.

*Location:***Zoom.us**: if you want to participate please contact organizers*Sakae Fuchino (Kobe University)*

Speaker:Speaker:

*Title*: A/the (possible) solution of the Continuum Problem*Abstract*. In this talk, I examine the following trichotomy which holds under the requirement that a sufficiently strong natural reflection principle should hold:The continuum (\(=2^{\aleph_0}\)) is either 1. \(\aleph_1\) or 2. \(\aleph_2\) or 3. fairly large.

Here, the fair largeness of the continuum can be expressed either in terms of weak mahloness and/or some other ``large'' cardinal notions compatible with the continuum, or even in terms of existence of some saturated ideals.

The reflection principles we consider here can be formulated as the following type of Downward Löwenheim-Skolem Theorems:

1'. For any structure A of countable signature, there is an elementary substructure B of A of cardinality \(<\aleph_2\) in terms of stationary logic.

2'. For any structure A of countable signature, there is an elementary substructure B of A of cardinality \(<2^{\aleph_0}\) in terms of stationary logic but only for formulas without free second order variables.

3'. For any structure A of countable signature, there is an elementary substructure B of A of cardinality \(<2^{\aleph_0}\) in terms of PKL logic (a variant of the stationary logic) in weak interpretation.

The reflection points \(<\aleph_2\) and \(<2^{\aleph_0}\) can be considered to be natural/necessary since the reflection down to \(<\aleph_2\) declares that \(\aleph_1\) strongly represents the situation of uncountability; the reflection down to \(<2^{\aleph_0}\) can be interpreted in the way that the reflection manifests that the continuum is very "rich".

The Downward Löwenheim-Skolem Theorems in terms of stationary logics can be also regarded as very natural principles: They can be characterized in terms of Diagonal Reflection Principles of Sean Cox.

Analyzing these three scenarios, we obtain the notion of Laver-generically large cardinals.

Existence of a Laver-generically supercompact cardinal

1''. for \(\sigma\)-closed pos implies 1'.;

2''. for proper pos implies 2'.; while the existence of a Laver-generically supercompact cardinal

3''. for ccc pos implies 3'.

The symmetry of the arguments involved suggests the possibility that the trichotomy might be a set-theoretic multiversal necessity.

If time allows, I shall also discuss about the reflection of non-metrizability of topological spaces, Rado's Conjecture and Galvin's Conjecture in connection with the reflection properties in 1., 2. and 3.

Most of the results to be presented here are obtained in a joint work with Hiroshi Sakai and André Ottenbreit Maschio Rodrigues.

# Gianluca Basso: The universal minimal flow of topological groups beyond Polish

11/06/20 22:59

Tuesday, June 16, 2020 17:15

*Location:***Zoom.us**: if you want to participate please contact organizers*Gianluca Basso (Université de Lausanne & Torino)*

Speaker:Speaker:

*Title*: The universal minimal flow of topological groups beyond Polish*Abstract*. When \(G\) is a Polish group, one way of knowing that it has "nice" dynamics is to show that \(M(G)\), the universal minimal flow of \(G\), is metrizable. For non-Polish groups, this is not the relevant dividing line: the universal minimal flow of \( \mathrm{Sym}(\kappa) \) is the space of linear orders on \(\kappa\)—not a metrizable space, but still "nice"—, for example. In this talk, we present a set of equivalent properties of topological groups which characterize having "nice" dynamics. We show that the class of groups satisfying such properties is closed under some topological operations and use this to compute the universal minimal flows of some concrete groups, like \(\mathrm{Homeo}(\omega_{1})\). This is joint work with Andy Zucker.# Wiesław Kubiś: Uniform homogeneity

03/06/20 20:45

Tuesday, June 9, 2020 17:15

*Location:***Zoom.us**: if you want to participate please contact organizers*Wiesław Kubiś (Czech Academy of Sciences)*

Speaker:Speaker:

*Title*: Uniform homogeneity*Abstract*. A mathematical structure is called homogeneous if every isomorphism between its small substructures extends to an automorphism. Typically, "small" means "finite" or "finitely generated". A stronger variant, which we call "uniform homogeneity" requires that for each small substructure there is a suitable extension operator. We shall present examples of homogeneous but uniformly homogeneous structures. The talk is based on two works: one joint with S. Shelah (https://arxiv.org/abs/1811.09650), another one joint with B. Kuzeljevic (https://arxiv.org/abs/2004.13643).