Large cardinals as principles of Structural Reflection

TL;DR

This paper introduces new structural reflection principles in set theory, demonstrating their equivalence to large cardinal axioms and thus providing a philosophical and mathematical foundation for their naturalness.

Contribution

It develops novel principles of Structural Reflection and proves their equivalence to established large cardinal axioms across the hierarchy.

Findings

Structural Reflection principles align with large cardinal axioms.

Different forms of Structural Reflection cover all large cardinal levels.

The principles support the naturalness of large cardinals in set theory.

Abstract

After discussing the limitations inherent to all set-theoretic reflection principles akin to those studied by A. L\'evy et. al. in the 1960's, we introduce new principles of reflection based on the general notion of \emph{Structural Reflection} and argue that they are in strong agreement with the conception of reflection implicit in Cantor's original idea of the unknowability of the \emph{Absolute}, which was subsequently developed in the works of Ackermann, L\'evy, G\"odel, Reinhardt, and others. We then present a comprehensive survey of results showing that different forms of the new principles of Structural Reflection are equivalent to well-known large cardinals axioms covering all regions of the large-cardinal hierarchy, thereby justifying the naturalness of the latter.