On [formula omitted]-amorphous sets

Abstract

We study a notion of ‘ o-amorphous’ (in set theory without the axiom of choice) which bears the same relationship to ‘ o-minimal’ as ‘amorphous’ (studied in Truss, Ann. Pure Appl. Logic 73 (1995) 191–233) does to ‘strongly minimal’. A linearly ordered set is said to be o-amorphous if its only subsets are finite unions of intervals. This turns out to be a relatively straightforward case, and we can provide a complete ‘classification’, subject to the same provisos as in Truss (1995). The reason is that since o-amorphous is an essentially second-order notion, it corresponds more accurately to ℵ 0 - categorical o-minimal, and our classification is thus very similar to the one given in (Pillay and Steinhorn, Trans. Amer. Math. Soc. 295 (1986) 565–592) for that case. More interesting structures arise if we replace ‘interval’ in the definition by ‘convex set’, giving us the class of weakly o-amorphous sets. Here, in fact, there are so many examples that a complete classification seems out of the question. We illustrate some of the structures which these may exhibit, and classify them in certain instances not too far removed from the o-amorphous case.