Intrinsic Justification for Large Cardinals and Structural Reflection

Abstract

We deal with the complex issue of whether large cardinals are intrinsically justified principles of set theory (we call this the Intrinsicness Issue). In order to do this, we review, in a systematic fashion, (1.) the abstract principles that have been formulated to motivate them, as well as (2.) their mathematical expressions, and assess the justifiability of both on the grounds of the (iterative) concept of set. A parallel, but closely linked, issue is whether there exist mathematical principles able to yield all known large cardinals (we call this the Universality Issue), and we also test principles for their responses to this issue. Finally, we discuss the first author's Structural Reflection Principles (SRPs), and their response to Intrinsicness and Universality. We conclude the paper with some considerations on the global justifiability of SRPs, and on alternative construals of the concept of set also potentially able to intrinsically justify large cardinals