The consequent most immediate challenge in Geometry is to correct Euclid by establishing the theorems of Book VI on the basis of the methods and techniques of Books I-IV, avoiding the use of the ratio concept introduced in Book V . For Arithmetic the corresponding challenge is to establish the results of Book VII-IX without resort to the sort of iterative procedure that Euclid allows himself in the definition of multiplication . ( Book VII, Definition 15 . ) For Cantorian Arithmetic the main challenge would be to show that the great body of infinitary mathematics-the disciplines flowing in one way or another from the calculus-does not require unbounded quantifiers and consequently that the instances of the Replacement Schema of the Zermelo-Fraenkel axioms for set theory involving such quantifiers, are, as well as being disallowed by Mayberry s general philosophy, in any case technically redundant.
