Philip Ehrlich
Philosophy Department
Ohio University

"All Numbers Great and Small"

ABSTRACT: In his monograph On Numbers and Games , J. H. Conway introduced a real-closed field containing the reals and the ordinals as well as a great many other numbers including 1/omega , omega/2 and omega - pi 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-Goedel set theory, it may be said to contain "All Numbers Great and Small." In this respect, bears the same relation to ordered fields that the ordered field of real numbers bears to Archimedean ordered fields.

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 I have called it, 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, it being understood that x is simpler than y just in case x is a predecessor of y in the tree.

My talk will provide an introduction to Conway's remarkable theory, a brief overview of some of my recent contributions thereto, and a discussion of why people with philosophical interests should care about it.

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