# 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.