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ON EXISTENCE IN SET THEORY, PART III: APPLICATIONS TO NEW AXIOMS

The present paper applies the conceptual apparatus developed in [1] and [2] to new axioms, and devises new tools for an analysis of the search for new axioms. Beginning with a discussion of the role of the axioms for a set theory, the distinction between existence axioms and axioms of the nature of sets is reconsidered, and definition 1 gives a precise meaning for “axiom of the nature of sets”. After this preliminary discussion, the present paper treats the comparison among new axioms, and among collections of new axioms, from the point of view of existence. The conceptual apparatus is then applied to analyzing the theory obtained from ZFC by adding the axiom “there is a measurable cardinal”, and the axiom V = L[U ], where L[U ] is the unique inner model of measurability associated with the first measurable cardinal, if