A CLASS OF 'NEAR-FINITE' ORDER-TYPES by JOHN L. HICKMAN in Canberra (Austra1ia)l) 1. Although in the formulations and proofs of the results presented hereunder we have had Zermelo-Fraenkel (ZF) set theory in mind, any other "reasonable" set theory without Choice would do just as well.
Most of our notation is standard; any non-standard terminology will be explained a t its first appearance. I n general we use upper-case Latin letters for sets, lower-case Latin letters for the corresponding elements, lower-case Greek letters for order-types, and upper-case Greek letters for classes of order-types. Ordinals are assumed to be defined in such a manner that each is the set of all preceding ordinals. We say that a set X is finite if there is an injection f : X + n for some natural number n, and that X is Dedekind-finite if for every injection f : X + X we have f"X = 9.
An infinite Dedekind-finite set is called medial.
If X is an ordered set, then X is called well-ordered if every non-empty subset of X has a minimal element, and doubly-well-ordered if every non-empty subset of X has both a minimal and a maximal element. We generalize these notions by saying that the (ordered) set X is general-well-ordered if there is no increasing injection f : w -+ X , where w is (and throughout this paper will always be) the first transfinite ordinal, and that X is g-finite if there is neither an increasing injection f : w --+ X nor a decreasing injection f : w -+ X . Within ZF we can show that every finite set is Dedekind-finite, every wellordered set iq general-well-ordered, and that every doubly-well-ordered set is g-finite; if we are prepared to accept a little help from the Axiom of Choice, then we can also prove the converses of the.;e three statements.
In a previous paper we looked at general-well-ordered (gwo) sets, and in this present paper we wish to examine g-finite sets, paying particular attention to the arithmetical properties of their order-types (g-numbers). As a start, we make the easily checked but useful observation that a set is g-finite if and only if it is ordered and Dedekind-finite. Since it is well-known that there are ZF-models containing ordered medial sets, we see that the class of g-numbers is not necessarily identical with the class of natural numbers. Furthermore, from the fact that the class of Dedekind-finite sets is closed under unions and Cartesian products, we conclude that the class of g-numbers is closed under addition and multiplication. We shall always denote this class by r.
A map f : X + Y where X , Y are ordered sets is called a rnorphism if f is orderpreserving and f"X is an initial segment of Y . It was shown in [l] that if X , Y are gwo, then there is at most one morphism X + Y , and since every g-finite set is gwo, the same result holds for g-finite sets. However, since the proof of this is very simple and the result very important, we outline the proof of the g-finite case below.