Amorphous set

In set theory, an amorphous set is an infinite set which is not the disjoint union of two infinite subsets.[1]

Existence

Amorphous sets cannot exist if the axiom of choice is assumed. However, a model of Zermelo–Fraenkel set theory (without choice) in which an amorphous set exists was given by Azriel Lévy, using an earlier model of Andrzej Mostowski.[2]

Additional properties

Every amorphous set is Dedekind-finite, meaning that is has no bijection to a proper subset of itself. To see this, suppose that S is a set that does have a bijection f to a proper subset. For each i ≥ 0 define Si to be the set of elements that belong to the image of the i-fold composition of f with itself but not to the image of the (i + 1)-fold composition. Then each Si is non-empty, so the union of the sets Si with even indices would be an infinite set whose complement is also infinite, showing that S cannot be amorphous. However, the converse is not necessarily true: it is consistent for there to exist Dedekind-finite sets that are not amorphous.[2]

No amorphous set can be linearly ordered.[3][4] Because the image of an amorphous set is itself either amorphous or finite, it follows that every function from an amorphous set to a linearly ordered set has only a finite image.

The cofinite filter on an amorphous set is an ultrafilter.

Variations

If π is a partition of an amorphous set into finite subsets, then there must be exactly one integer n(π) such that π has infinitely many subsets of size n; for, if every size was used finitely many times, or if more than one size was used infinitely many times, this information could be used to coarsen the partition and split π into two infinite subsets. If an amorphous set has the additional property that, for every partition π, n(π) = 1, then it is called strictly amorphous, and if there is a finite upper bound on n(π) then the set is called bounded amorphous. It is consistent with ZF that amorphous sets exist and are all bounded, or that they exist and are all unbounded.[1]

References

  1. 1 2 Truss, J. K. (1995), "The structure of amorphous sets", Annals of Pure and Applied Logic, 73 (2): 191–233, doi:10.1016/0168-0072(94)00024-W, MR 1332569.
  2. 1 2 Lévy, A. (1958), "The independence of various definitions of finiteness" (PDF), Fundamenta Mathematicae, 46: 1–13, MR 0098671.
  3. ↑ Truss, John (1974), "Classes of Dedekind finite cardinals" (PDF), Fundamenta Mathematicae, 84 (3): 187–208, MR 0469760.
  4. ↑ de la Cruz, Omar; Dzhafarov, Damir D.; Hall, Eric J. (2006), "Definitions of finiteness based on order properties" (PDF), Fundamenta Mathematicae, 189 (2): 155–172, doi:10.4064/fm189-2-5, MR 2214576. In particular this is the combination of the implications Ia → II → Δ3 which de la Cruz et al. credit respectively to Lévy (1958) and Truss (1974).
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