On o-amorphous sets

The first-order structure of weakly Dedekind-finite set

“…The similarity between o-amorphous sets and Ko-categorical o-minimal theories was studied extensively in [3]. While we cannot add much to those results, we will show that the methods developed above can be used to simplify some of the proofs.…”

“…The notions in Definitions 2.5 and 2.5 were first given in [3], and are inspired by model theoretic notions.…”

“…The proof is similar to that of Theorem 2.4. Thus, we can immediately deduce from [13] that, for example, no o-amorphous set can carry a group structure, and we can obtain the complete classification of o-amorphous sets in [3] from that for Ko-categorical o-minimal sets in [13].…”

“…There is a corresponding relation between weakly o-amorphous sets (as defined in [3]) and weakly o-minimal theories (a definition and discussion of which can be found in [6]). which may be given by the methods above.…”

“…Hitherto, the structures of Dedekind-finite sets have been explored in a direct fashion, by testing the properties of the sets by methods of ad hoc model construction or by direct proofs that carefully avoid the use of the Axiom of Choice (AC). Examples of such approaches can be found in, for example, [16,9,3].…”