The finite and the infinite
type: Item
instance of: scholarly article
title: The finite and the infinite
zbMATH Open document ID: 1236.18004
author: Horst Herrlich
published in: Applied Categorical Structures
publication date: 2011-05-11T00:00:00Z
review text: The paper analyses the relation between various finiteness classes in ZF set theory without Choice. It starts with an an extensive background collecting citations of the most influential mathematicians and philosophers (Weyl, Hilbert, Russell, Pascal, Galileo, Leibniz, Bolzano, Dedekind, von Neumann) on the nature of infinity. Then, the author introduces a notion of finiteness class. This is a class of sets that contains all finite sets (i.e., sets that embed into the set \(\mathbb N\) of natural numbers), does not contain \(\mathbb N\), and is hereditary (in the sense that together with any set \(X\) it contains all sets that embed injectively into \(X\)). In Theorems 9 and 10, the author studies the relation between five finiteness classes: finite sets; A-finite sets (those that cannot be written as the disjoint union of two infinite sets); B-finite sets (those that contain no infinite linearly orderable subset); C-finite sets (those that cannot be mapped onto \(\mathbb N\)); and D-finite sets (those that contain no copy of \(\mathbb N\)). It is shown that the classes of finite sets and of D-finite sets are the smallest and the largest finiteness classes, respectively. In Theorem 12, sixteen conditions equivalent to the existence of a unique finiteness class are given. In Theorem 22 the author compares the five finiteness classes by their properties: coheredity; (finite) summability; (finite) productivity; power-stability. The paper ends with three open problems. The first of them asks whether each C-finite set is B-finite.
reviewed by: Taras Banakh
Mathematics Subject Classification ID: 18B05
Mathematics Subject Classification ID: 03E25
zbMATH DE Number: 5886919
zbMATH Keywords: axiom of choice
zbMATH Keywords: finiteness class
zbMATH Keywords: finite
zbMATH Keywords: Dedekind-finite
zbMATH Keywords: hereditary
zbMATH Keywords: cohereditary
zbMATH Keywords: summable
zbMATH Keywords: productive
zbMATH Keywords: power-closed
zbMATH Keywords: full subcategories of set
MaRDI profile type: Publication
full work available at URL: s10485-009-9212-5
OpenAlex ID: W2053878336
cites work: The shrinking principle and the axiom of choice
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cites work: Axiom of choice
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cites work: Definitions of finite
cites work: The axiom of choice
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DOI: 10.1007/S10485-009-9212-5
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description: scientific article; zbMATH DE number 5886919