Which axioms of set theory are intrinsically justified?

TL;DR

This paper explores the justification of certain set-theoretic axioms through reflection principles and large cardinal hypotheses, establishing new connections and implications among them.

Contribution

It analyzes the relationships between various reflection principles and large cardinals, showing how strengthened principles imply the existence of supercompact and huge cardinals.

Findings

A natural strengthening of Roberts' reflection principle yields supercompact cardinals.

The work resolves open questions about the strength of Marshall's theories.

It demonstrates equivalences between certain reflection principles and large cardinal axioms.

Abstract

We recently formulated a new large-cardinal axiom of strength intermediate between a totally indescribable cardinal and an $\omega$-Erd\H{o}s cardinal, positing the existence of what we called an "extremely reflective cardinal", and we showed that the property of being extremely reflective was in fact equivalent to the property of being remarkable, and we sought to argue that this axiom should be seen as intrinsically justified. This built on related earlier work in which the notion of an $\alpha$-reflective cardinal was formulated. Then Welch and Roberts put forward a family of reflection principles, Welch's principle implying the existence of a proper class of Shelah cardinals and provably consistent relative to a superstrong cardinal, and Roberts' principle implying the existence of a proper class of 1-extendible cardinals and provably consistent relative to a 2-extendible cardinal. Roberts tentatively argued that his principle should be seen as intrinsically justified (at least on the assumption that a weaker form of reflection involving reflection of second-order formulas with a second-order parameter should be seen as intrinsically justified). This work overlapped with previous work of Victoria Marshall's on reflection principles. We analyze the relationship between reflection principles equivalent to those studied in my earlier work and stronger but similar reflection principles which are natural extensions of those of Welch and Roberts. We also show how a natural strengthening of Roberts' reflection principle yields the existence of supercompact cardinals, and in the process solve a question which Marshall left open, of whether her theory $B_0(V_0)$ is strong enough to imply the existence of supercompact cardinals. We also manage to resolve negatively her question of whether her theory $B_1(V_0)$ implies the existence of a huge cardinal.