The naïve idea of “size” for collections seems to obey both Aristotle’s Principle: “the whole is greater than its parts” and Cantor’s Principle: “1-to-1 correspondences preserve size”. Notoriously, Aristotle’s and Cantor’s principles are incompatible for infinite collections. Cantor’s theory of cardinalities weakens the former principle to “the part is not greater than the whole”, but the outcoming cardinal arithmetic is very unusual. It does not allow for inverse operations, and so there is no direct way of introducing infinitesimal numbers. Here (...) we maintain Aristotle’s principle, instead halving Cantor’s principle to “equinumerous collections are in 1–1 correspondence”. In this way we obtain a very nice arithmetic: in fact, our “numerosities” may be taken to be nonstandard integers. These numerosities appear naturally suited to sets of ordinals, but they depend, for generic sets, on a “labelling” of the universe by ordinals. The problem of finding a canonical way of attaching numerosities to all sets seems to be worth further investigation. (shrink)

It is well known that the validity of Choice Principles is problematic in non-standard Set Theories which do not abide by the Limitation of Size Principle. In this paper we discuss the consistency of various Choice Principles with respect to the Generalized Positive Comprehension Principle . The Principle GPC allows to take as sets those classes which can be specified by Generalized Positive Formulae, e.g. the universe. In particular we give a complete characterization of which choice principles hold in Hyperuniverses. (...) Hyperuniverses are structures which arose independently in Non-well-founded Set Theory and in Mathematical Semantics of Concurrent Programming Languages and are hitherto the only existing models of GPC. Hyperuniverses are naturally endowed with a κ-compact uniform κ-topology and are uniformly isomorphic to their exponential space, i.e. the space of their closed subsets endowed with the Exponential Uniformity. (shrink)

The proof of Lemma 5 in our paper “Choice Principles in Hyperuniverses” [3], contains an error. In the present note we show that the statement of that lemma is false and hence the Axiom of Choice fails in all κ-hyperuniverses, for uncountable κ. However, a weaker version of Lemma 5 can be proved, which implies that the Linear Ordering Principle holds in all κ-metric κ-hyperuniverses.