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

THE JOURNAL OF SYMBOLIC LOGIC 

Volume 66, Number 3, Sept. 2001 

NUMBER SYSTEMS WITH SIMPLICITY HIERARCHIES: 

A GENERALIZATION OF CONWAY'S THEORY OF SURREAL 

NUMBERS 

PHILIP EHRLICH 

Introduction. In his monograph On Numbers 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 -co, co/2, 1/co, wS and co - 7T 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 Neumann- 

Bernays-Godel set theory with Global Choice, henceforth NBG [cf. 21, Ch. 4], it may 

be said to contain "All Numbers Great and Small." In this respect, No 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 homogeneous universal 

Archimedean orderedfield, No is (up to isomorphism) the unique homogeneous 

universal orderedfield [14]; also see [10], [12], [13]. 1 

However, in addition to its distinguished structure as an ordered field, No 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 simplicity 

hierarchy, 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. 

1 For the purpose of this paper, an ordered field (Archimedean ordered field) A is said to be homogeneous 

universal if it is universal every ordered field (Archimedean ordered field) whose universe is 

a set or a proper class of NBG can be embedded in A and it is homogeneous 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 [10], [12], [13] and [14] the terms absolutely homogeneous universal, absolutely universal, and absolutely 

homogeneous were respectively employed in their steads. 

(?) 2001, Association for Symbolic Logic 

0022-4812/01/6603-001 5/$3.80 

1231

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 with 

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 Numbers 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


DEFINITION 1. A lexicographically ordered binary tree is an ordered binary tree 

(A,


either z E Rs(x) 0 Ls(y) or z E Rs(y) 0 Ls(x) depending upon whether x


DEFINITION 2. (A, +,


PROPOSITION 1. (i) 0 is the simplest element of (as well as the unique root in) an 

s-hierarchical ordered group A, i.e., 0 ={0 o 0}A; (ii) for each element x of an s- 

hierarchical ordered group A, -x = {-xR I XL}A; (iii) 1 is the simplest positive 

element of an s-hierarchical orderedfield A, i.e., 1 ={0 I}A. 

1.3. Complete s-hierarchical ordered algebraic structures. A binary tree will be 

said to be full if every element has two immediate successors and every chain of 

infinite limit length < On has an immediate successor. As Theorem 4 below makes 

clear, in the case of lexicographically ordered binary trees the concept of a full 

binary tree is intimately related to the following conception. 

DEFINITION 6. A lexicographically ordered binary tree (A,


XL, XR, YL, and yRrange over the members of Ls(x), Rs(x), Ls(Y), and Rs(Y), respectively. 

DEFINITION OF X + Y. 

DEFINITION OF-X.4 

DEFINITION OF Xy. 

x + y {X +LYy,X+ YL XR+ yX+yR}. 

-X = {xR I -X L} 

xy {xLy + xyLXLyLxRy + xyR R R y 

x y + xy -x YRXY + xyL _xRyL}. 

1.4. s-Hierarchical embeddings and complete s-hierarchical ordered algebraic structures. 

The present subsection culminates in a theorem that indicates that (No,


PA' (f (x)). Conversely, if f is a mapping of the said kind and x E A, then 

f (Ls(x))


existence of f (x) so defined and thereby, in virtue of Lemma 2, the universal nature 

of A if A is an s-hierarchical ordered group or an s-hierarchical ordered field. For 

the case where A is an s-hierarchical ordered vector space, the universality of A is 

established by appealing to Lemma 2 in conjunction with the following result whose 

simple proof is left to the reader: 

PROPOSITION. Let K be an s-hierarchical orderedfield, f be the unique s-hierarchical 

mappingfrom K to No, and No be the multiplication in the s-hierarchical orderedfield 

No. (No, +,


?2. The spectrum of s-hierarchical ordered vector spaces. 

2.1. s-Hierarchical ordered vector spaces. There are a great many ordered vector 

spaces that are not isomorphic to initial subspaces of No. The present subsection 

culminates with a result specifying necessary and sufficient conditions for those 

which are. To prepare the way for our proof of the said theorem, we first establish 

two preliminary results, the proof of the first of which makes use of the following 

ELEMENTARY OBSERVATION. If A is an initial subtree of a lexicographically ordered 

binary tree A' and x ={L R}A', where (L, R) is a partition of A, then A U {x} is 

an initial subtree of A'. 

PROOF. If a C prA (x), then Ls(a) C Ls(x) < {x} < Rs(x) D Rs(a). But x - 

{L I R}A; and so, by Theorem 2, there is a z c L U R such that z > each x' c LS W 

and z < each x' c Rs(x), which with Theorem 2 implies a


obtain x = rb + a 

{L(x) I R(x) } where 

L(x) = rLb + (rbL -r LbL + a) rb + a , rR b + (rbR -r RbR + a) }, 

R(x) {rLb + (rbR - rLbR + a) rb + aR, rRb + (rbR -r RbL + a) }, 

and rL, rR, bL, bR, aL and aR are typical members of Ls(I.), Rs(I.), Ls(b), Rs(b), Ls(a) 

and Rs(a), respectively But since A is a vector space over K, (rbL - rLbL + a), 

(rbR- rRbR + a), (rbR- rLbR + a) and (rbR - rRbL + a) are all in A, since 

r, rL, rR c K and a, bL, bR C A; and consequently, rLb + (rbL -rLbL + a), 

r b+ (rbRrRbR + a), rL b + (rbR -rLbR + a) and rRb + (rbR rRbL a) 

are all in A< since each such sum is in one or another A(YE) where y < f,. Moreover, 

sums of the forms rb + a L and rb + a R are in A< since each such sum is in one or 

another A(p,,) where a < v; and as such L(x) U R(x) C A


ordered class A we mean a pair of nonempty subclasses X and Y of A where X has 

no greatest member, Y has no least member, X U Y = A, and X < Y. 

DEFINITION 10. Let R = D U {{X I Y}: (X, Y) is a Dedekind gap in ED} where 

D = {x c No: PNo (x) < co}; as such, R is the set of all members of Uf


on these structures by identifying them with isomorphic copies of particular formal 

power series fields. One of the central components in our proof of the identification 

is the relation that exists between the formal power series fields in question and 

the complete s-hierarchical ordered field (No, +H, *H


Lead (A) nA' ifA' is an initial subgroup of A; (vii) ifA' is an initial subgroup of A, then 

(Lead (A'),


DEFINITION 12. For each y c No, ogy = {, noLs(Y) | oy) }NO 

THEOREM 12. coy = ID-1 (y) = t0,n(L | 21nw(R} for each y C No and each 

ordered pair (L, R) such that y = {L I R}INo. 

Lsl(cl') = o{WX: x C Ls(y)} and Rsl(,Y) = fox x C Rs(y)}. 

And so, since (D-1 is s-hierarchical, 

PROOF. Plainly, coo = ID7o (0) = { 0}. Now suppose y c No - {0} and cox 

iD-1 (x) for all x c Ls(y) U Rs(y). Then, since by Theorems 10(v) and 11 

(D~~ 1y ~-1 

ONo () 

~D'():xCL~~ 

{No (X :t~)}|{No 

(- x 

()XC s(y)}} 

we have 

LeadNo 

-i 

CL 

(y) - { CD |Rs y) }Lead(No) 

And so, by Theorem 10 (iii) and the fact that {0} U {nwoLs(y) } is cofinal with L*y) 

and { 1oRs(y) } is coinitial with Rs*l, 

we have 

(D (y) {0, ncowLs(Y) |1 CoRs(y) } cy 

But, by Theorem 2, if y = {L IR}N, then L is cofinal with Ls(Y) and R is coinitial 

with Rs(Y), from which it follows that { 0, nL} is cofinal with {0, nwLs(y) } and 

{?w1n~)R} is coinitial with 

{ 

{0, nw L} < {WCoY} < { (OR R 

_)R5'(Y)}. But then coy = {0,nwOL (OR}N, since 

Since No is a vector space over R and every member of No - {0} is Archimedean 

equivalent to exactly one member of {coy: y C No} C No+, a familiar classical 

argument (cf. [19, Lemma 2.4]) leads to the following 

SIMPLE RESULT. If x c No - {0}, then there is a unique r C R - {O} and a unique 

y c No for which Ix - rcoy

Z 


PROOF. If x c No - {0}, there is a unique ro c R - {0} and a unique yo C No for 

which Ix - (s0 + rocoy)I


simple consequence of Theorem 13 together with Definition 13 makes clear, the use 

of the summation and product signs are as appropriate as they are revealing. 

BASIC OBSERVATION. coy . r = rowy (i.e., = r No coY) for each y C No and each 

r c R - {fO}, andfor each nonempty descending sequence (yO4f


by their respective Conway names is inspired by a result due to Conway. Here 

and henceforth, for the sake of descriptive simplicity, we permit the concept of the 

Conway name of a surreal number to be naturally expanded so as to allow for the 

insertion and deletion of "dummy" terms with zeros for coefficients. 

THEOREM 16. (No,+H, H


x+HYis an approximation of (x +H y)+Hw. But then x +Hy


PROOF. Let A be an initial subfield of No. To prove the "only if" portion of 

the theorem it suffices to show A is an approximation complete subfield of No 

(see Definition 14) where G = {y: coy E Lead (A)} is an initial subgroup of No 

for then plainly { E< rtY- c R (G)0,1 : w


({bu} U U6


PROOF. Let A be an ordered field whose universe may be a proper class, K G, ,


The nature of the system of omnific integers of a nontrivial initial subgroup of 

No is greatly clarified by the following theorem, the nonroutine portion of which 

follows immediately from Conway's proof of Theorem 31 of [7]. 

THEOREM 20. If A is a nontrivial initial subgroup of No, then Oz (A) is the subclass 

of all x c A such that 

x 

Zcow . ac, 

a

apt . 


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 

E09c 

*oa c aa 

a


It is evident that every s-hierarchical ordered group is a-Archimedean for some 

a. Also evident is 

THEOREM 24. Let A be a nontrivial s-hierarchical ordered group and h: 

On (A) -> On be an s-hierarchical mapping. A is a-Archimedean if and only iffor 

each a E A there is an q Ec On (A) such that -a < a < q where h (q) < a. Moreover, 

if A is an s-hierarchical ordered field, then A is a-Archimedean if and only iffor all 

a,b E A+ where a > b there is an q < a such that qb > a; furthermore, A is 

Archimedean if and only if A is co-Archimedean. 

By appealing to Theorem 23 together with the definitions of +H and H as 

stated in Theorem 16, it is easy to see that an initial sequence {a,: a < Pl < On} 

of ordinals is closed under addition (multiplication) if and only if P = w0 for 

some ordinal (indecomposable ordinal) p < On. Accordingly, if for each ordinal 

(indecomposable ordinal) p < On we let SG (aLP) (SR (ai)) be the ordered class 

of all ordinals less than w0 with sums and products defined as in Theorem 16, then 

the SG (aiP)s (SR (0i)s) so defined are the only subsemigroups (subsemirings) of 

No that consist of initial sequences of ordinals of No.8 And this together with 

Theorems 21, 23 and 24 and the fact that for each s-hierarchical ordered structure 

A there is one and only one s-hierarchical mapping hA: A -* No, hA being an 

s-hierarchical embedding, we have 

THEOREM 25. Every nontrivial s-hierarchical ordered group (s-hierarchical ordered 

field) A is w0-Archimedean for some nonzero ordinal (nonzero indecomposable ordinal) 

p < On; and if A is an w0-Archimedean s-hierarchical ordered group (shierarchical 

ordered field) then A contains a cofinal, canonical subsemigroup (subsemiring) 

On (A) called the ordinal part of A that is isomorphic to SG (00) 

(SR (0P)) and which consists of all a Ec A such that a = {L I 0}A for some 

L C A; On (A) in turn is contained in a discrete, canonical subgroup (subring) 

Oz (A) of A called the omnific integer part of A consisting of all x E A such that 

x = {x - e I x + e}A where e is the simplest positive element of A; in addition, for 

each x E A there is a z E Oz (A) such that z < x < z + e.9 

?6. Concluding remarks. While laying the groundwork, the present paper merely 

takes a first step in the development of a general theory of s-hierarchical ordered algebraic 

structures. We conclude by mentioning a few directions for further research 

that the author (and, hopefully, others) will turn to in the not too distant future. To 

begin with, in addition to shedding further light on the nature of the s-hierarchical 

8As the reader will notice, for each ordinal A, SG (co9') is essentially the ordered semigroup of all 

ordinals (written in Cantor normal form) less than cow with order defined by first differences and 

addition defined a la Hessenberg [cf. 27], and for each indecomposable ordinal A, SR (cow') is essentially 

the ordered semiring that results from supplementing SG (co9') with a Hessenberg product [cf. 27]. 

Unlike the SG (co9')s, thus construed, which have been discussed in the literature [26], the SR (co9')s 

appear to be new (except for the cases where the cow s are regular initial numbers [27]). 

9The omnific integer part of an s-hierarchical ordered field is an integer part in the sense of Mourgues 

and Ressayre [23], i.e., a discrete subring I of an ordered field A in which for each x E A there is a z E I 

such that z < x < z + 1. However, while every s-hierarchical ordered field has an integer part, there are 

ordered fields having integer parts that do not admit relational extensions to an s-hierarchical ordered 

field each Hahn field JR (G) where G does not admit a relational extension to an s-hierarchical ordered 

group is an example of such a system.


ordered algebraic structures we have investigated thus far, it remains to investigate 

their natural generalizations the s-hierarchical ordered algebraic structures that 

are defined by replacing the references to ordered groups, ordered fields and ordered 

K-vector spaces in Definitions 2-4 with references to ordered semigroups, ordered 

rings, and ordered K-modules, respectively. While some of our results such as 

Lemma 2 and Theorem 5 as well as their proofs carry over to these more general 

classes of s-hierarchical ordered structures, their isolation and analysis remain wide 

open. Similar remarks apply to s-hierarchical analogs of ordered semirings and ordered 

K-semimodules, which also deserve attention. In addition, unlike the ordered 

fields of reals and surreals which admit (up to isomorphism) only one relational 

extension to an s-hierarchical structure, many s-hierarchable structures admit relational 

extensions to nonisomorphic s-hierarchical structures and an analysis of the 

classes of such alternative extensions also remains to be provided. Finally, while we 

have devoted the bulk of our attention to s-hierarchical ordered structures per se, 

detailed analyses of the hierarchical relations that exist between their respective elements 

also remain to be provided including a detailed analysis of the tree-theoretic 

relations that exist between surreal numbers denoted by their respective Conway 

names. 

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DEPARTMENT 

OHIO UNIVERSITY 

OF PHILOSOPHY 

ATHENS, OH 45701, USA 

E-mail: ehrlich~ohiou.edu

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