Large Cardinals as Principles of Structural Reflection

Abstract

After discussing the limitations inherent to all settheoretic reflection principles akin to those studied by A. Lévy et. al. in the 1960's, we introduce new principles of reflection based on the general notion of 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 Absolute, which was subsequently developed in the works of Ackermann, Lévy, Gödel, 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.