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We recently formulated a new large-cardinal axiom of strength intermediatebetween 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 factequivalent to the property of being remarkable, and we sought to argue thatthis axiom should be seen as intrinsically justified. This built on relatedearlier work in which the notion of an $\alpha$-reflective cardinal wasformulated. Then Welch and Roberts put forward a family of reflectionprinciples, Welch's principle implying the existence of a proper class ofShelah cardinals and provably consistent relative to a superstrong cardinal,and Roberts' principle implying the existence of a proper class of 1-extendiblecardinals and provably consistent relative to a 2-extendible cardinal. Robertstentatively argued that his principle should be seen as intrinsically justified(at least on the assumption that a weaker form of reflection involvingreflection of second-order formulas with a second-order parameter should beseen as intrinsically justified). This work overlapped with previous work ofVictoria Marshall's on reflection principles. We analyze the relationshipbetween reflection principles equivalent to those studied in my earlier workand stronger but similar reflection principles which are natural extensions ofthose of Welch and Roberts. We also show how a natural strengthening ofRoberts' reflection principle yields the existence of supercompact cardinals,and in the process solve a question which Marshall left open, of whether hertheory $B_0(V_0)$ is strong enough to imply the existence of supercompactcardinals. We also manage to resolve negatively her question of whether hertheory $B_1(V_0)$ implies the existence of a huge cardinal.

Which axioms of set theory are intrinsically justified?