Bounded Symbiosis and Upwards Reflection
Lorenzo Galeotti, Yurii Khomskii, Jouko Väänänen

Abstract:
Bagaria and V\"a\"an\"anen developed a framework for studying the large cardinal strength of \emph{downwards} L\"owenheim-Skolem theorems and related set theoretic reflection properties. The main tool was the notion of \emph{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.