1. If A is strongly amorphous , then its power set P is dually Dedekind infinite, i. e., every function from P onto P is injective. 2. The class of “inexhaustible” sets is not closed under supersets unless AC holds.

A set is said to be amorphous if it is infinite, but is not the disjoint union of two infinite subsets. Thus amorphous sets can exist only if the axiom of choice is false. We give a general study of the structure which an amorphous set can carry, with the object of eventually obtaining a complete classification. The principal types of amorphous set we distinguish are the following: amorphous sets not of projective type, either bounded or unbounded size of members (...) of partitions of the set into finite pieces), and amorphous sets of projective type, meaning that the set admits a non-degenerate pregeometry, over finite fields either of bounded cardinality or of unbounded cardinality. The hope is that all amorphous sets will be of one of these types. Examples of each sort are constructed, and a reconstruction result for bounded amorphous sets is presented, indicating that the amorphous sets of this kind constructed in the paper are the only possible ones. The final section examines some questions concerned with the resulting cardinal arithmetic. (shrink)

In [3] F. P. Ramsey proved as a theorem of Zermelo-Fraenkel set theory (ZF) with the Axiom of Choice (AC) the following result:(1) Theorem. Let A be an infinite class. For each integer n and partition {X, Y} of the size n subsets of A, there exists an infinite subclass of A all of whose size n subsets are contained in only one of X or Y.

I wish to thank Klaus Kühnle who streamlined in [8] several of my definitions and proofs concerning the subject matter of this paper. Some ideas and results arose from discussions with Klaus Leeb. Jan Johannsen discovered some mistakes in an earlier version.