Intrinsic Justification for Large Cardinals and Structural Reflection

TL;DR

This paper examines whether large cardinals are intrinsically justified by set theory principles, analyzing their motivations, mathematical expressions, and the role of Structural Reflection Principles in addressing these foundational issues.

Contribution

It provides a systematic review of principles motivating large cardinals, assesses their justifiability, and explores the role of Structural Reflection Principles in intrinsic justification.

Findings

Structural Reflection Principles offer a promising response to intrinsic and universal justifications.

The paper identifies limitations of existing principles in fully justifying large cardinals.

Alternative conceptions of set may also intrinsically justify large cardinals.

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.