Quasi-selective ultrafilters and asymptotic numerosities

DOI10.1016/J.AIM.2012.06.021zbMATH Open1270.03105arXiv1011.2089OpenAlexW2089421517MaRDI QIDQ456765

Mauro Di Nasso, Marco Forti, Andreas Blass

Publication date: 16 October 2012

Published in: Advances in Mathematics (Search for Journal in Brave)

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.

Full work available at URL: https://arxiv.org/abs/1011.2089

zbMATH Keywords

ultrafiltersemiringhypernatural numbersnonstandard modelnumerosity

Mathematics Subject Classification ID

Other combinatorial set theory (03E05) Other set-theoretic hypotheses and axioms (03E65) Ultraproducts and related constructions (03C20)

Cites Work

• Model theory.
• The Shelah P-point independence theorem
• Numerosities of labelled sets: A new way of counting
• An Aristotelian notion of size
• Numerosities of point sets over the real line
• Combinatorial Cardinal Characteristics of the Continuum
• Fine asymptotic densities for sets of natural numbers
• Weakly Ramsey P Points
• Ultrafilter Mappings and Their Dedekind Cuts
• Ultrafilters on a countable set
• Title not available (Why is that?)
• Title not available (Why is that?)
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Related Items (13)

One-dimensional packing: maximality implies rationality ⋮ SLOW P-POINT ULTRAFILTERS ⋮ Generic existence of interval P-points ⋮ EUCLIDEAN NUMBERS AND NUMEROSITIES ⋮ On embedding certain partial orders into the P-points under Rudin-Keisler and Tukey reducibility ⋮ Rapid interval P-points ⋮ A note on extensions of asymptotic density ⋮ Title not available (Why is that?) ⋮ SET SIZE AND THE PART–WHOLE PRINCIPLE ⋮ Germ order for one-dimensional packings ⋮ Numerosities of point sets over the real line ⋮ Title not available (Why is that?) ⋮ Natural numerosities of sets of tuples

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