The structure of amorphous sets
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 equivalent to a Fraenkel-Mostowski model \({\mathcal M}_G\) which is generated by the wreath product \(G \text{Wr SF} (\omega)\), where \(G\) is a finite transitive permutation group.
scientific article; zbMATH DE number 768742
scientific article