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The construction of groups in models of set theory that fail the Axiom of Choice

Published online by Cambridge University Press:  17 April 2009

J.L. Hickman
J.L. Hickman
Affiliation:
Department of Mathematics, Institute of Advanced Studies, Australian National University, Canberra, ACT.
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Abstract

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The purpose of this paper is to show that a well-known method for constructing “queer” sets in models of ZF set theory is also applicable to certain algebraic structures. An infinite set is called “quasi-minimal” if every subset of it is either finite or cofinite. In Section 1 I set out the two systems of set theory to be used in this paper, and illustrate the technique in its most fundamental form by constructing a model of set theory containing a quasi-minimal set. In Section 2 I show that by choosing the parameters appropriately, one can use this technique to obtain models of set theory containing groups whose carriers are quasi-minimal. In the third section various independence results are deduced from the existence of such models: in particular, it is shown that it is possible in ZF set theory to have an infinite group that satisfies both the ascending and descending chain conditions. The quasi-minimal groups constructed in Section 2 were all elementary abelian; in Section 4 it is shown that this was not just chance, but that in fact all quasi-minimal groups must be of this type. Finally in Section 5 permutations and permutation groups on quasi-minimal sets are examined.

Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 14 , Issue 2 , April 1976 , pp. 199 - 232
Copyright
Copyright © Australian Mathematical Society 1976

References

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