Abstract

Novel (categorical) axiomatizations of the classical arithmetic and geometric continua are provided and it is noted that by simply deleting the Archimedean condition one obtains (categorical) axiomatizations of J.H. Conway's ordered field No and its elementary n-dimensional metric Euclidean, hyperbolic and elliptic geometric counterparts. On the basis of this and related considerations it is suggested that whereas the classical arithmetic and geometric continua should merely be regarded as arithmetic and geometric continua modulo the Archimedean condition, No and its geometric counterparts may be regarded as absolute arithmetic and geometric continua modulo von Neumann-Bernays-Godel set theory.