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Conway introduced a real-closed field containing the reals and the ordinals as well as a great many less familiar numbers including \u2212\u03c9, \u03c9\/2, 1\/\u03c9,<jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S1079898600000226_inline1\"\/>and \u03c9 \u2212 \u03c0 to name only a few. Indeed, this particular real-closed field, which Conway calls<jats:bold>No<\/jats:bold>, is so remarkably inclusive that, subject to the proviso that numbers\u2014construed here as members of ordered fields\u2014be individually definable in terms of sets of NBG (von Neumann\u2013Bernays\u2013G\u00f6del set theory with global choice), it may be said to contain \u201cAll Numbers Great and Small.\u201d In this respect,<jats:bold>No<\/jats:bold>bears much the same relation to ordered fields that the system \u211d of real numbers bears to Archimedean ordered fields.<\/jats:p><jats:p>In Part I of the present paper, we suggest that whereas \u211d should merely be regarded as constituting an arithmetic continuum (modulo the Archimedean axiom),<jats:bold>No<\/jats:bold>may be regarded as a sort of absolute arithmetic continuum (modulo NBG), and in Part II we draw attention to the unifying framework<jats:bold>No<\/jats:bold>provides not only for the reals and the ordinals but also for an array of non-Archimedean ordered number systems that have arisen in connection with the theories of non-Archimedean ordered algebraic and geometric systems, the theory of the rate of growth of real functions and nonstandard analysis.<\/jats:p><jats:p>In addition to its inclusive structure as an ordered field, the system<jats:bold>No<\/jats:bold>of surreal numbers has a rich algebraico-tree-theoretic structure\u2014a simplicity hierarchical structure\u2014that emerges from the recursive clauses in terms of which it is defined. In the development of<jats:bold>No<\/jats:bold>outlined in the present paper, in which the surreals emerge vis-\u00e0-vis a generalization of the von Neumann ordinal construction, the simplicity hierarchical features of<jats:bold>No<\/jats:bold>are brought to the fore and play central roles in the aforementioned unification of systems of numbers great and small and in some of the more revealing characterizations of<jats:bold>No<\/jats:bold>as an absolute continuum.<\/jats:p>","DOI":"10.2178\/bsl\/1327328438","type":"journal-article","created":{"date-parts":[[2012,1,23]],"date-time":"2012-01-23T15:38:21Z","timestamp":1327333101000},"page":"1-45","source":"Crossref","is-referenced-by-count":44,"title":["The Absolute Arithmetic Continuum and the Unification Of all Numbers Great and Small"],"prefix":"10.1017","volume":"18","author":[{"given":"Philip","family":"Ehrlich","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"56","published-online":{"date-parts":[[2014,1,15]]},"reference":[{"key":"S1079898600000226_ref061","volume-title":"Rings of continuous functions","author":"Gillman","year":"1961"},{"key":"S1079898600000226_ref122","doi-asserted-by":"publisher","DOI":"10.4064\/fm167-2-3"},{"key":"S1079898600000226_ref121","unstructured":"van den Dries Lou [1991], Truncation of power series (remarks on a manuscript by Ressayre), unpublished."},{"key":"S1079898600000226_ref103","first-page":"739","article-title":"Alcune osservazioni sugli ordini d'infinito delle funzioni","volume":"5","author":"Pincherle","year":"1884","journal-title":"Memorie della R. 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