between subfields of A whose universes are sets can be extended to an ... number of types of s-hierarchical ordered structures-groups, fields, vector spaces- as well as a .... only if x and y have a common predecessor z such that x < z < y. ...... is said to be injinitesimal (in absolute value) relative to b; the class of all members.
THEJOURUAL OT SY\IBOLIC LOGIC Volume 66. Number 3. Sept. 2001
NUMBER SYSTEMS WITH SIMPLICITY HIERARCHIES:
A GENERALIZATION O F CONWAY'S THEORY O F SURREAL
NUMBERS
PHILIP EHRLICH
Introduction. In his monograph On Nur?zbers and Games [7]. J. H. Conway introduced a real-closed field containing the reals and the ordinals as well as a great many other numbers including -w, w/2. l / w , fi and w - 71 to name only a few. Indeed, this particular real-closed field. which Conway calls No, is so remarkably inclusive that, subject to the proviso that numbers-construed here as members of ordered "number" fields-be individually definable in terms of sets of von Neur?zanrzBernays-Godelset theory with Global Choice. henceforth NBG [cf. 21. Ch. 41, it may be said to contain "All Numbers Great and Small." In this respect, N o bears much the same relation to ordered fields that the system of real numbers bears to Archimedean ordered fields. This can be made precise by saying that whereas the ordered field of reals is (up to isomorphism) the unique Izornogeneous universal Archimedean ordered$eld, N o is (up to isomorphism) the unique homogeneous universal orderedfield [14]; also see [lo], [12], [13]. However, in addition to its distinguished structure as an ordered field. N o has a rich hierarchical structure that (implicitly) emerges from the recursive clauses in terms of which it is defined. This algebraico-tree-theoretic structure, or sin~plicity Izierarchy. as we have called it [15], depends upon No's (implicit) structure as a lexicographically ordered binary tree and arises from the fact that the sums and products of any two members of the tree are the simplest possible elements of the tree consistent with No's structure as an ordered group and an ordered field. respectively.
'
Received September 2. 1998; revised March 29, 2000. Portions of this paper were presented at the 1998 ASL Spring Meeting in Los Angeles, the 1998 ASL Summer Meeting in Prague. the 1999 Mal'tsev Meeting in Novosibirsk, and the University of Notre Dame Mathematical Logic Seminar. Research supported by the National Science Foundation (Scholars Award # SBR 9602154) and Ohio University. The author wishes to express his thanks to these institutions for their support and to Lou van den Dries and the referee for suggesting helpful ways for streamlining and improving the exposition. For the purpose of this paper. an ordered field (Archimedean ordered field) A is said to be Itornogeneous universal if it is universal-every ordered field (Archimedean ordered field) whose universe is a set or a proper class of NBG call be embedded in A-and it is lto~itogeneous-every isomorphism between subfields of A whose universes are sets can be extended to an automorphism of A. Since model theorists frequently use the above italicized terms in more general senses. in the model-theoretic settings of [lo], [12]. [I31 and [14] the terms absolutely hoinogeneozts unive~sal.absolutely unive~sal.and irbsolutely honzogeiteous were respectively employed in their steads. @ 2001. Association for Syinbolic Logic
0022-4812/01/6603-0015/$3.80
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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-lzierarchical ordered structures as we call them. In 5 1 we define a number of types of s-hierarchical ordered structures-groups, fields, vector spacesas well as a corresponding type of s-lzierarchical 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 nlapping 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 with the completeness of No enables us to characterize No. up to isomorphism, as the unique complete as well as the unique rzonextensible and the unique universal, s-hierarchical ordered group (s-hierarchical ordered field: s-hierarchical ordered vector space). Following this, in 52 and 54 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 Numbers in January of 1989. I. Every divisible orderedabelian group is isor?zorplzicto an initialsubgroup of No. 11. Every real-closed ordered$eld is isomorphic to an initial subfield of No. In §3; as part of the groundwork for the proof of 11;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 R (No),,. 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 2 ~ o r specifically. e 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 I]). 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].
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(subring) Oz(A) of A-the on~nijicinteger part of A-in which for each x E A there x < z e where e-which is the simplest positive is a z E Oz(A) such that z 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 on~nij?cintegers, 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 0 1 7 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 < p 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,,,p (R, x { a ) ) . Tuples involving proper classes, more generally, are likewise understood.
b and rrlb > n .
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The nature of the system of on~nificintegers of a nontrivial initial subgroup of No is greatly clarified by the following theorem, the nonroutine portion of which follows iillinediately from Conway's proof of Theorem 31 of [7]. THEOREM 20. Ij"A is a nontrivial initial subgroup of No, then Oz ( A ) is the subclass of all x E A such that
wlzere is a (possibly empty) descending sequence of nonnegative members of No and a, is a nonzero integer if y, = 0; and so. i f A is a nontrivial initial subgroup (initial subfield) oj No. then Oz ( A ) is a discrete subgroup (subring) of A in whichfor eachx E A tlzereisaz E O z ( A ) such thatz 5 x < z 1.
+
DEFINITION 18. An element a of an initial subtree A of No will be said to be an ordinal if a has a representation of the form a = {L 1 0) (i.e., if and only if a is a (possibly empty) sequence of +s). By On ( A ) we mean the subclass of all ordinals of A. THEOREM 21. On (A)-which i~ an initial subtree, as well as a cofinal subclass, of c1as.s in which for each a E On ( A ) . a = {ordinals y < a / ) :i f A-is a ~t~ell-ordered A = No, then for every subset S of On ( A ) there a member of On ( A ) greater tlzan every nqeniber of S : if A is a nontrivial s-hierarchical ordered group (s-hierarchical orderedfield), then On ( A ) is a subsen~igroup(subsemiring) of O z ( A ) . PROOF.The assertion between the dashes is obvious, and the remainder of the first two parts of the theorem follows from an argument of Conway [7, p. 281. Moreover, if a. p E On ( A ) where A is a nontrivial initial subgroup of No, then a - 1 each ordinal y < a and a + 1 5 each y E 0. which, together with the fact that a = {ordinals ;-< a 1) . implies that a = { a - 1 / a 1)" and, hence, A that On ( A ) C_ Oz ( A ) . Finally, since a = { a L 1) " and /3 = { P L } , a P = E 0 1 7 ( A ) and. if A is an s-hierarchical ordered field, in which { a L+ p, a + PL
>
+
+
i case ap is well defined. aP = {aLP aPL - aLPL1) " t On ( A ) . In virtue of Theorem 21 there is a one-to-one order preserving mapping between No's ordinals and the ordinals defined in any of the more usual fashions. This being the case. we may adopt the convention that identifies the class On of all ordinals (that are sets) with the class On ( N o ) rather than with the class of von Neumann ordinals as we have heretofore supposed. With this convention henceforth assumed to be in place. the familiar Cantorian operations on ordinals-written a t , P , a .,, P , and a"p-as well as the following classical definitions and results based thereon are understood to be formulated in terms of the members of On = On ( N o ) . Moreover. for the sake of convenience. henceforth we will extend the use of the term 'ordinal' to include On = (x,),,~,, where x, = + for all a E On. extend the ordering on On to On U {On) in the expected manner and. on occasion. write On = won, the expression on the right being the "Conway name" of the "leader" (0. noL\lon, 1) in No u {On). As the reader will recall. an ordinal number a is said to be (additively) indecomposable if P I,y.< a whenever P.;) < a . Moreover. if 0 < a < On. then a is indecoillposable if and only if a = w ' l v for soille ordinal number cp < On. This
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together with other familiar theorems concerning ordinals leads to the following well-known theorem due to Cantor: Every ordinal a < On has a unique Cantor Normal Form. i.e., a unique representation of the form
where (cp,)n 0, and the 'c' affixed to the summation sign indicates the sum is Cantorian. By extending Cantorian exponentiation of ordinals to the indecomposable ordinal On. it is easy to see that On = o'< On and, hence, that w'' On = won. The following theorem, which is central to the remainder of the paper. demonstrates that the theory of Conway names generalizes the theory of Cantor normal forms for classical ordinals as well. The proof makes tacit use of the well-known result that to compare ordinals written in Cantor normal form one compares them by first differences [e.g.. 20, p. 1271. ., a, = C,,, o'+'.. a, whenever (cp,) n 0 and suppose the result holds for all y < z. Since z > 0, z = ., a, for some m < w ; and ., a, = EL
