Quasi-selective ultrafilters and asymptotic numerosities

TL;DR

This paper introduces quasi-selective ultrafilters, a new class between selective ultrafilters and P-points, and explores their connection to asymptotic numerosities, which extend finite cardinalities to countable sets with a structured algebraic framework.

Contribution

It defines quasi-selective ultrafilters and establishes their equivalence with the existence of asymptotic numerosities for all sets of tuples of natural numbers.

Findings

Quasi-selective ultrafilters are intermediate between selective ultrafilters and P-points.

Existence of quasi-selective ultrafilters is equivalent to the existence of asymptotic numerosities.

Asymptotic numerosities form a hypernatural number system extending finite cardinalities.

Abstract

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 all sets of tuples of natural numbers. Such numerosities are hypernatural numbers that generalize finite cardinalities 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 sequences of tuples of natural numbers.