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The notion of infinite measuring number and its relevance in the intuition of infinity

By David Tall


In this paper a concept of infinity is described which extrapolates themeasuring properties of number rather thancounting aspects (which lead to cardinal number theory).\ud Infinite measuring numbers are part of a coherent number system extending the real numbers, including both infinitely large and infinitely small quantities. A suitable extension is the superreal number system described here; an alternative extension is the hyperreal number field used in non-standard analysis which is also mentioned.\ud Various theorems are proved in careful detail to illustrate that certain properties of infinity which might be considered false in a cardinal sense are true in a measuring sense. Thus cardinal infinity is now only one of a choice of possible extensions of the number concept to the infinite case. It is therefore inappropriate to judge the correctness of intuitions of infinity within a cardinal framework alone, especially those intuitions which relate to measurement rather than one-one correspondence.\ud The same comments apply in general to the analysis of naive intuitions within an extrapolated formal framework.\u

Topics: QA
Publisher: Springer Netherlands
Year: 1980
OAI identifier: oai:wrap.warwick.ac.uk:506

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