Conway's Theory Of Surreal Numbers

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\textbf{Number Systems with Simplicity Hierarchies: A Generalization of

Conway's Theory Of Surreal Numbers}

Philip Ehrlich

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In his monograph \textit{On Numbers and Games,} J. H. Conway introduced an

ordered field $No$ consisting, as he aptly quips, of ``all numbers great and

small.'' In addition to its distinguished structure as a maximally inclusive

ordered field, however, $No$ has a rich algebraico-tree-theoretic structure,

or \textit{simplicity hierarchy,} that emerges from the recursive clauses in

terms of which it is defined. In the present paper, we introduce a novel

class of structures whose properties generalize those of $No$ and explore

some of the relations that exist between $No$ and this more general class of

\textit{s-hierarchical ordered structures} as we call them. More

specifically, we define a number of types of s-hierarchical ordered

structures (groups; fields; etc.) as well as a corresponding type of\textit{%

\ s-hierarchical mapping,} identify $No$ as a \textit{complete}

s-hierarchical ordered group (ordered field; etc.), 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 monomorphisms of their respective

kinds (ordered groups; ordered fields; etc.) whose images are initial

subtrees of $No$. This together with the completeness of $No$ enables us to

characterize $No$, up to isomorphism, as the unique complete as well as the

unique \textit{universal} and the unique\textit{\ non-extensible},

s-hierarchical ordered group (field, etc.). Following this, we begin the

process of uncovering the spectrum of s-hierarchical ordered structures.

Given the nature of $No$ alluded to above, this reduces to revealing the

spectrum of \textit{s-hierarchical substructures} of $No$, i.e., the

subgroups, subfields, etc. of $No$ that are initial subtrees of $No$. Among

our findings are: \textit{every divisible ordered abelian group is

isomorphic to an s-hierarchical subgroup of} $No$, and \textit{every

real-closed ordered field is isomorphic to an s-hierarchical subfield of} $No

$. We also generalize Conway's theories of \textit{ordinals} and \textit{%

omnific integers} by showing that every s-hierarchical ordered field $A$

contains a cofinal, canonical subsemiring $On(A)$ - the\textit{\ ordinal

part }of $A$ - which in turn is contained in a discrete, canonical subring $%

Oz(A)$ of $No$ - the\textit{\ omnific integer part }of $A$ - in which for

each $x\in A-Oz(A)$ there is a $y\in Oz(A)$ such that $y

. When $A$ is a substructure of $No$, $Oz(A)$ is a subring of $No$'s omnific

integers and $On(A)$ is a subsemiring of $No$'s subsemiring of all ordinals

(with sums and products defined naturally).