Amorphous set

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. This is already sufficient for proving the consistency of the existence of an amorphous set with Zermelo–Fraenkel set theory with atoms. After Cohen's initial work on forcing in 1963, proofs of the consistency of amorphous sets with Zermelo–Fraenkel set theory were obtained.

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 S is a set that does have a bijection f to a proper subset. For each natural number i\ge 0 define S_i 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 S_i is non-empty, so the union of the sets S_i with even indices would be an infinite set whose complement in S is also infinite, showing that 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.

No amorphous set can be linearly ordered.{{citation

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 \Pi is a partition of an amorphous set into finite subsets, then there must be exactly one integer n(\Pi) such that \Pi has infinitely many subsets of size n. This is because, 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 \Pi into two infinite subsets. If an amorphous set has the additional property that, for every partition \Pi, n(\Pi)=1, then it is called strictly amorphous or strongly amorphous, and if there is a finite upper bound on 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.

References

References

1. Jech, Thomas J.. (2008). "The axiom of choice". Dover Publications.
2. Plotkin, Jacob Manuel. (November 1969). "Generic Embeddings". [[The Journal of Symbolic Logic]].
3. Lévy, A.. (1958). "The independence of various definitions of finiteness". [[Fundamenta Mathematicae]].