Bounded symbiosis and upwards reflection

Abstract In Bagaria (J Symb Log 81(2), 584–604, 2016), Bagaria and Väänänen developed a framework for studying the large cardinal strength of downwards Löwenheim-Skolem theorems and related set theoretic reflection properties. The main tool was the notion of symbiosis , originally introduced by the third author in Väänänen (Applications of set theory to generalized quantifiers. PhD thesis, University of Manchester, 1967); Väänänen (in Logic Colloquium ’78 (Mons, 1978), volume 97 of Stud. Logic Foundations Math., pages 391–421. North-Holland, Amsterdam 1979) 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 Löwenheim-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 Löwenheim–Skolem-type principles for second order logic.