Quasi-selective ultrafilters and asymptotic numerosities

We isolate a new class of ultrafilters on N, called "quasi-selective" because they are intermediate between selective ultrafilters and P-points. (Under the Continuum Hypothesis these three classes are distinct.) The existence of quasi-selective ultrafilters is equivalent to the existence of "asymptotic numerosities" for tinsels of tuples A subset of N-k. Such numerosities are hypernatural numbers that generalize finite carclinalities to countable point sets. Most notably, they maintain the structure of ordered semiring, and, in a precise sense, they allow for a natural extension of asymptotic density to all sets of tuples of natural numbers. (C) 2012 Elsevier Inc. All rights reserved.