Continuity
In the decades bracketing the turn of the twentieth century, the real number system was dubbed the arithmetic continuum because it was held that this number system is completely adequate for the analytic representation of all types of continuous phenomena. In accordance with this view, the geometric linear continuum is assumed to be isomorphic with the arithmetic continuum, the axioms of geometry being so selected to ensure this would be the case. In honor of Georg Cantor (1845–1918) and Richard Dedekind (1831–1916), who first proposed this mathematico-philosophical thesis, the presumed correspondence between the two structures is sometimes called theCantor-Dedekind axiom.
Since their appearance, the late nineteenth-century constructions of the real numbers have undergone set-theoretical and logical refinement, and the systems of rational and integer numbers on which they are based have themselves been given a set-theoretic foundation. During this period the Cantor-Dedekind philosophy of the continuum also emerged as a pillar of standard mathematical philosophy that underlies the standard formulation of analysis, the standard analytic and synthetic theories of the geometrical linear continuum, and the standard axiomatic theories of continuous magnitude more generally.
Since its inception, however, there has never been a time at which the Cantor-Dedekind philosophy has either met with universal acceptance or has been without competitors.
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