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. Fraenkel constructed a permutation model of Zermelo–Fraenkel with Atoms in which the set of atoms is an amorphous set. ^[2] After Cohen's initial work on forcing in 1963, proofs of the consistency of amorphous sets with Zermelo–Fraenkel were obtained. ^[3]

Additional properties

Every amorphous set is Dedekind-finite, meaning that it has no bijection to a proper subset of itself. To see this, suppose that ${\displaystyle S}$ is a set that does have a bijection ${\displaystyle f}$ to a proper subset. For each natural number ${\displaystyle i\geq 0}$ define ${\displaystyle S_{i}}$ to be the set of elements that belong to the image of the ${\displaystyle i}$-fold composition of f with itself but not to the image of the ${\displaystyle (i+1)}$-fold composition. Then each ${\displaystyle S_{i}}$ is non-empty, so the union of the sets ${\displaystyle S_{i}}$ with even indices would be an infinite set whose complement in ${\displaystyle S}$ is also infinite, showing that ${\displaystyle S}$ cannot be amorphous. However, the converse is not necessarily true: it is consistent for there to exist infinite Dedekind-finite sets that are not amorphous. ^[4]

No amorphous set can be linearly ordered. ^{[5] [6]} 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. This is because the complement of each infinite subset must not be infinite, so every subset is either finite or cofinite.

Variations

If ${\displaystyle \Pi }$ is a partition of an amorphous set into finite subsets, then there must be exactly one integer ${\displaystyle n(\Pi )}$ such that ${\displaystyle \Pi }$ has infinitely many subsets of size ${\displaystyle n}$; 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 ${\displaystyle \Pi }$ into two infinite subsets. If an amorphous set has the additional property that, for every partition ${\displaystyle \Pi }$, ${\displaystyle n(\Pi )=1}$, then it is called strictly amorphous or strongly amorphous, and if there is a finite upper bound on ${\displaystyle n(\Pi )}$ 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. 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. Jech, Thomas J. (2008), The axiom of choice, Mineola, N.Y.: Dover Publications, ISBN   978-0486318257, OCLC   761390829
3. Plotkin, Jacob Manuel (November 1969), "Generic Embeddings", The Journal of Symbolic Logic, 34 (3): 388–394, doi:10.2307/2270904, ISSN   0022-4812, JSTOR   2270904, MR   0252211, S2CID   250347797
4. Lévy, A. (1958), "The independence of various definitions of finiteness" (PDF), Fundamenta Mathematicae, 46: 1–13, doi:10.4064/fm-46-1-1-13, MR   0098671 .
5. Truss, John (1974), "Classes of Dedekind finite cardinals" (PDF), Fundamenta Mathematicae, 84 (3): 187–208, doi:10.4064/fm-84-3-187-208, MR   0469760 .
6. 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 ${\displaystyle {\text{Ia}}\Rightarrow {\text{II}}\Rightarrow \Delta _{3}}$ which de la Cruz et al. credit respectively to Lévy (1958) and Truss (1974).