The deductive relationships between six statements are examined in set theory without the axiom of choice. Each of these statements follows from the axiom of choice and involves linear orderings in some way.

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 set theory without the Axiom of Choice math formula, we study the deductive strength of the statements math formula, math formula, math formula, and math formula. Among various results, we prove that none of the above statements is provable without using some form of choice, math formula is equivalent to math formula, math formula + math formula implies math formula, math formula does not imply math formula in math formula, math formula does not imply math formula in math formula (...) and math formula is strictly weaker than math formula in math formula. (shrink)