Risk and theoretical equivalence in mathematical foundations

Consistency, interpretability and probability are three key instruments in the mathematical philosopher's kit when it comes to questions of foundational theory comparison. This paper aims to bring these tools together with a focus on theories capable of providing foundations for mathematics with a particular emphasis on set theory. A number of counterintuitive results emerge which are then addressed by offering a novel framework based on what we call pointwise interpretability. We then investigate a plausible, existing instance of this framework, the generic multiverse, and demonstrate that it can be naturally situated within our pointwise interpretability framework.

Automated summary

Consistency, interpretability, and probability are vital tools for mathematicians when comparing foundational theories. This paper focuses on theories that establish foundations for mathematics, particularly in set theory. It introduces a new framework called pointwise interpretability to address counterintuitive results and explores the application of this framework in the generic multiverse.