On quasi-amorphous sets

Abstract.

A set is said to be amorphous if it is infinite, but cannot be written as the disjoint union of two infinite sets. The possible structures which an amorphous set can carry were discussed in [5]. Here we study an analogous notion at the next level up, that is to say replacing finite/infinite by countable/uncountable, saying that a set is quasi-amorphous if it is uncountable, but is not the disjoint union of two uncountable sets, and every infinite subset has a countably infinite subset. We use the Fraenkel–Mostowski method to give many examples showing the diverse structures which can arise as quasi-amorphous sets, for instance carrying a projective geometry, or a linear ordering, or both; reconstruction results in the style of [1] are harder to come by in this case.