The structure of amorphous sets
type: Item
instance of: scholarly article
title: The structure of amorphous sets
zbMATH Open document ID: 0827.03030
DOI: 10.1016/0168-0072(94)00024-W
published in: Annals of Pure and Applied Logic
publication date: 1995-12-13T00:00:00Z
review text: In set theory without the axiom of choice the following definition is meaningful. An infinite set \(A\) is amorphous if its infinite subsets are cofinite. The author investigates Fraenkel-Mostowski permutation models whose set \(A\) of atoms is amorphous. They differ with respect to the locally finite geometries which are associated with \(A\). A geometry is induced in the canonical way from a closure operator \(f: e(A)\to e(A)\) on the set \(e(A)\) of all finite subsets of \(A\) which satisfies the exchange axiom [cf. \textit{M. Aigner}, Combinatorial theory (1979; Zbl 0415.05001)]. An amorphous set is of projective type if it carries a non-degenerate locally finite locally homogeneous geometry. A typical example is the set \(A\) of the atoms of the model \({\mathcal M}\) which is generated by the topological automorphism group of the countably dimensional projective geometry over a finite field \(F_q\). Although in this example \(F_q\) may be reconstructed from \(A\), the author constructs an amorphous set of projective type which admits geometries corresponding to different fields. In view of the first main result, Corollary 3.3., the characteristics of these fields coincide. In the degenerate case the geometry reduces to a partition of \(A\) into finite pieces. It admits a gauge \(n\geq 1\) such that all except finitely many equivalence classes have \(n\) elements. \(A\) is bounded amorphous if there is an upper bound on the possible gauges. If \(A\) is of projective type, then \(A\) is not bounded amorphous. An example of a bounded amorphous set with the bound 1 on the gauges is provided by the set \(A\) of the atoms of the basic Fraenkel model \({\mathcal M}_1\) which is generated by the group \(\text{SF} (\omega)\) of the permutations with finite supports and the topology of pointwise convergence. The second main result of the paper, Corollary 5.11, asserts that up to technical provisos a permutation model with a bounded amorphous set of atoms is elementarily…
Mathematics Subject Classification ID: 03E25
Mathematics Subject Classification ID: 03C45
zbMATH DE Number: 768742
zbMATH Keywords: strongly minimal set
zbMATH Keywords: set theory without the axiom of choice
zbMATH Keywords: Fraenkel- Mostowski permutation models
zbMATH Keywords: locally finite geometries
zbMATH Keywords: amorphous set
zbMATH Keywords: locally homogeneous geometry
zbMATH Keywords: projective geometry
zbMATH Keywords: partition
zbMATH Keywords: gauge
Wikidata QID: Q59702030
author: John K. Truss
reviewed by: Norbert Brunner
MaRDI profile type: Publication
cites work: What's so special about \((\mathbb{Z}/4\mathbb{Z})^ \omega\)?
cites work: On strongly minimal sets
cites work: Q3794175
cites work: \(\aleph _ 0\)-categorical, \(\aleph _ 0\)-stable structures
cites work: On quasi-amorphous sets
cites work: Homogeneous Geometries
cites work: Models of ZF-set theory
cites work: Ueber eine abgeschwaechte Fassung des Auswahlaxioms
cites work: Q5641139
cites work: Cohomology of Structures and Some Problems of Ahlbrandt and Ziegler
cites work: Lectures in set theory with particular emphasis on the method of forcing
cites work: Q5553271
cites work: The independence of various definitions of finiteness
cites work: Small sets with large power sets
cites work: Cancellation laws in the arithmetic of cardinals
cites work: Classes of Dedekind finite cardinals
Recommended article: On quasi-amorphous sets
Recommended article: On o-amorphous sets
Recommended article: Strongly Amorphous Sets and Dual Dedekind Infinity
Recommended article: On the structure of set-mappings
Recommended article: Q4296712
Recommended article: The ambient structure of basic sets
Recommended article: Q2941888
Recommended article: Syndetic sets and amenability
Recommended article: Structures coordinatized by indiscernible sets
Recommended article: Q3794175
description: scientific article; zbMATH DE number 768742