Bounded Symbiosis and Upwards Reflection
Abstract
Bagaria and V\"a\"an\"anen developed a framework for studying the large cardinal strength of downwards L\"owenheim-Skolem theorems and related set theoretic reflection properties. The main tool was the notion of symbiosis, originally introduced by the third author. Symbiosis provides a way of relating model theoretic properties of strong logics to definability in set theory. In this paper we continue the systematic investigation of symbiosis and apply it to upwards L\"owenheim-Skolem theorems and reflection principles. To achieve this, we need to adapt the notion of symbiosis to a new form, called bounded symbiosis. As one easy application, we obtain upper and lower bounds for the large cardinal strength of upwards L\"owenheim-Skolem-type principles for second order logic.
Cite
@article{arxiv.2007.14718,
title = {Bounded Symbiosis and Upwards Reflection},
author = {Lorenzo Galeotti and Yurii Khomskii and Jouko Väänänen},
journal= {arXiv preprint arXiv:2007.14718},
year = {2020}
}