Bounded Symbiosis and Upwards Reflection

TL;DR

This paper extends the concept of symbiosis to bounded symbiosis to analyze the large cardinal strength of upwards L"owenheim-Skolem theorems and reflection principles in set theory.

Contribution

It introduces bounded symbiosis and applies it to establish bounds on the large cardinal strength of upwards L"owenheim-Skolem principles for second order logic.

Findings

Established bounds for large cardinal strength of upwards L"owenheim-Skolem principles

Extended symbiosis framework to bounded symbiosis for new applications

Linked model theoretic properties to set-theoretic definability

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.