ON THE EXISTENCE OF REAL-CLOSED FIELDS THAT ARE ^-SETS OF POWER ««(x)

Introduction. In the theory of i7„-sets three main theorems stand out: I. An tya-set is universal for totally ordered sets of power not exceeding fc$„. II. Two r/a-sets of power tí« are isomorphic. III. If Na is regular and if ^2s0 then these three theorems hold for the category of totally ordered Abelian groups and order preserving (group) isomorphisms; the special group being totally ordered, Abelian, divisible, and an rça-set. Erdös, Gillman, and Henriksen [8] (see also Gillman and Jerison [ll]) proved that if a>0 then I and II hold for the category of totally ordered fields and order preserving (ring) isomorphisms; the special field being real-closed and an i7„-set. It was also shown in [8] that III holds for this category and special object if a= 1. However in case a> 1, III was left open both in [8] and in [11]. The initial aim of these researches was to show that, assuming a>0, Ha regular, and 2s0. Let PIG} denote the field of formal power series with exponents in G and coefficients in R, the reals. P{G} is an Tja-set but its power exceeds fc$«. Let P{G}a = {/£P{G} : the support of /is of power less than K„}. Then R{G}„, again a real-closed field, is an ?ja-set, and is of power Ha. The only difficult point in these verifications was the proof that P{g} and R {G} a are 7;a-sets. The proof arrived at by the author did not involve the multiplication in these fields, but depended wholly on their structure as a