Symbiosis and Compactness Properties
Jonathan Osinski
Abstract:
We investigate connections between model-theoretic properties of extensions of first-order logic and set-theoretic principles. We build on work of Bagaria and Väänänen, and of Galeotti, Khomskii and Väänänen, which used the notions of symbiosis and bounded symbiosis between a logic L and a predicate of set theory R, respectively, to show that if L and R are (boundedly) symbiotic, (upwards) Löwenheim-Skolem properties of L are equivalent to certain (upwards) reflection principles involving R. Similarly, we consider whether under the assumption of symbiosis compactness properties of L are related to some set-theoretic principle involving R.
For this purpose, we give a thorough introduction to symbiosis and the concepts from abstract model theory and set theory needed in its study. We further give a proof of a characterization of compactness properties of L in terms of extensions of specific partial orders stated by Väänänen. We use this and the novel concept of (R, κ)-extensions to formulate a set-theoretic principle which describes that in classes which are definable under the usage of R there exist (R, κ)-extensions with upper bounds for such partial orders. We show that this principle is related to compactness properties of a logic L symbiotic to R.