Parsons and Shapiro argue that the iterative conception of sets provides independent, non-circular grounds for accepting the cumulative hierarchy V before any axiomatization is formulated.

?Rate how convincing each reason is below to see the overall strength.

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Axiomatization(as in the title of Reichenbach's work): The process of taking a theory and organizing it into a set of basic assumptions (called axioms) from which everything else can be logically derived.
Cumulative hierarchy V(as the mathematical structure being justified): The standard mathematical picture of how all sets fit together, where simpler sets come first and more complex sets are built from them. Mathematicians use 'V' as shorthand for this organized structure.
Iterative conception of sets(as the theory Parsons and Shapiro defend): A way of thinking about sets (collections of objects) where you build them up step-by-step, starting with no objects, then making sets from those, then making sets from those sets, and so on forever. It's like building a tower where each level is made from the levels below it.
Non-circular grounds(as what was missing for choosing between different confirmation functions)

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