17 December 2020
Yurii Khomskii (Amsterdam & Hamburg)
Bounded Symbiosis and Upwards Reflection
In , Bagaria and Väänänen developed a framework
for studying the large cardinal strength of Löwenheim-Skolem
theorems of strong logics using the notion of Symbiosis (originally
introduced by Väänänen in ). Symbiosis provides a way
of relating model theoretic properties of strong logics to definability
in set theory. We continue the systematic investigation of Symbiosis and
apply it to upwards Löwenheim-Skolem theorems and upwards
reflection principles. To achieve this, the notion of Symbiosis is
adapted to what we call "Bounded Symbiosis". As an application, we
provide some upper and lower bounds for the large cardinal strength of
upwards Löwenheim-Skolem principles of second order logic.
This is joint work with Lorenzo Galeotti and Jouko
 Joan Bagaria and Jouko Väänänen, "On the Symbiosis
Between Model-Theoretic and Set-Theoretic Properties of Large
Cardinals", Journal of Symbolic Logic 81 (2), 584–604.
 Jouko Väänänen, "Abstract logic and set theory. I.
Definability." In: Logic Colloquium '78 (Mons, 1978), vol. 97 of Studies
in Logic and the Foundations of Mathematics (North Holland, 1979),