Number Systems with Simplicity Hierarchies: A ... - Ohio University

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1232 PHILIP EHRLICH it being understood that x is simpler than y just in case x is a predecessor of y in the tree. In [15], the just-described simplicity hierarchy was brought to the fore2 and made part of an algebraico-tree-theoretic definition of No. In the pages that follow, we introduce a novel class of structures whose properties generalize those of No so construed and explore some of the relations that exist between No and this more general class of s-hierarchical ordered structures as we call them. In ? 1 we define a number of types of s-hierarchical ordered structures groups, fields, vector spacesas well as a corresponding type of s-hierarchical mapping, identify No as a complete s-hierarchical ordered group (s-hierarchical ordered field; s-hierarchical ordered vector space), and show that there is one and only one s-hierarchical mapping of an s-hierarchical ordered structure into No (or any complete s-hierarchical ordered structure, more generally). These mappings are found to be embeddings of their respective kinds whose images are initial subtrees of No, and this together <strong>with</strong> the completeness of No enables us to characterize No, up to isomorphism, as the unique complete as well as the unique nonextensible and the unique universal, s-hierarchical ordered group (s-hierarchical ordered field; s-hierarchical ordered vector space). Following this, in ?2 and ?4 we turn our attention to uncovering the spectrum of s-hierarchical ordered structures. Given the nature of No alluded to above, this reduces to revealing the spectrum of initial substructures of No, i.e., the subgroups, subfields, subspaces of No (considered as an s-hierarchical ordered algebraic structure) that are initial subtrees of No. Included among our findings are the following two results that were originally stated as conjectures by the author at the AMS special session on Surreal <strong>Number</strong>s in January of 1989. I. Every divisible ordered abelian group is isomorphic to an initial subgroup of No. II. Every real-closed orderedfield is isomorphic to an initial subfield of No. In ?3, as part of the groundwork for the proof of II, we provide novel proofs that each surreal number x can be represented by a unique formal sum which may be treated as a canonicalproper name of x and the closely related fact that No considered as an ordered field is isomorphic to the formal power series field JR (No)On. In ?5, we generalize and amplify Conway's theories of ordinals and omnific integers by showing that every nontrivial s-hierarchical ordered group (s-hierarchical ordered field) A contains a cofinal, canonical subsemigroup (subsemiring) On(A) the ordinalpart of A which in turn is contained in a discrete, canonical subgroup 2More specifically, in [15], following a suggestion of Conway, the just-described simplicity hierarchy was brought to the fore and thereby freed from the ambiguity that befalls it in Conway's own treatment in [7]. The ambiguity arises because remarks made in [7] make it possible (if not more likely) to interpret "x is simpler than y" as x has an earlier birthday than y, rather than in the manner specified above as Conway had intended (Private Conversation: see [15, pp. 257-258: note 1]). For some purposes the ambiguity is of little consequence. For example, one may show that No has precisely one automorphism that preserves simplicity regardless of which one of the above two interpretations of the simpler than relation is adopted [4], [5], [15]. On the other hand, as the succeeding pages only begin to show, from the standpoint of exploring the internal structure of No, it the tree-theoretic interpretation that is the more revealing. For treatments of No in which "x is simpler than y" is interpreted as x has an earlier birthday than y, see, for example, [4], [5], and [6].
NUMBER SYSTEMS WITH SIMPLICITY HIERARCHIES 1233 (subring) Oz(A) of A the omnific integer part of A-in which for each x E A there is a z E Oz(A) such that z < x < z + e where e which is the simplest positive element of A is the least positive element of Oz(A). When A is a substructure of No, e is the surreal number 1 and the members of On(A) and Oz(A) are called ordinals and omnific integers, respectively Finally, in ?6 we specify directions for further research. Throughout the paper the underlying set theory is assumed to be NBG and as such by class we mean set or proper class, the latter of which, in virtue of the Axiom of Global Choice, always has the "cardinality" of the class On of all ordinals. Moreover, since the usual definition of a sequence is not a legitimate conception in NBG when proper classes are involved, we follow the standard practice of understanding by a "structure" whose universe A is a proper class and whose finitary relations R,, 0 < a < fi E On, on A are classes (which may be operations or distinguished elements treated as special relations) the class (A x {0}) U R where R = Uo
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Number Systems with Simplicity Hierarchies: A ... - Ohio University

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