On forms of justification in set theory

In the contemporary philosophy of set theory, discussion of new axiomsthat purport to resolve independence necessitates an explanation of howthey come to bejustified. Ordinarily, justification is divided into two broadkinds:intrinsicjustification relates to how ‘intuitively plausible’ an axiomis, whereasextrinsicjustification supports an axiom by identifying certain‘desirable’ consequences. This paper puts pressure on how this distinctionis formulated and construed. In particular, we argue that the distinction asoften presented is neitherwell-demarcatednor sufficientlyprecise. Instead, wesuggest that the process of justification in set theory should not be thoughtof as neatly divisible in this way, but should rather be understood as a con-ceptually indivisible notion linked to the goal ofexplanation.