Strong downward L\"owenheim-Skolem theorems for stationary logics, III -- mixed support iteration

TL;DR

This paper investigates reflection principles related to stationary logic's L"owenheim-Skolem theorems, focusing on models built via mixed support iteration of supercompact cardinals and their impact on topological space properties.

Contribution

It extends previous work by analyzing models obtained through mixed support iteration, revealing independence results concerning non-metrizability reflection in topological spaces.

Findings

Reflection down to < 2^{}} of non-metrizability is independent of studied reflection properties.

Models constructed via mixed support iteration exhibit specific reflection behaviors.

The results connect large cardinal assumptions with topological and logical reflection principles.

Abstract

Continuing [Fuchino, Ottenbreit and Sakai[9, 10]] and [Fuchino and Ottenbreit[11]], we further study reflection principles in connection with the L\"owenheim-Skolem Theorems of stationary logics. In this paper, we mainly analyze the situations in the models obtained by mixed support iteration of a supercompact length and then collapsing another supercompact cardinal to make it $(2^{\aleph_0})^+$. We show, among other things, that the reflection down to $< 2^{\aleph_0}$ of the non-metrizability of topological spaces with small character is independent from the reflection properties studied in [Fuchino, Ottenbreit and Sakai[9, 10]] and [Fuchino and Ottenbreit[11]].