Abstract
(Non)denumerability The focus of this article is the rise of modern set theory which, according to Meschkowski, coincides with the first proof given in 1874 by Cantor of the non-denumerability of the real numbers. Later on he developed his well-known diagonal proof, which occupies a central position in this article. The argument of this article is directed towards the implicit supposition of the diagonal proof, to wit the acceptance of the actual infinite (preferably designated as the at once infinite). Without this assumption no conclusion to non-denumerability is possible. Various mathematicians and mathematical traditions of the twentieth century questioned the use of the actual infinite. A closer investigation is conducted in respect of two opponents of the actual infinite, namely Kaufmann and Wolff. The circular reasoning contained in their approach is highlighted and as alternative a non-circular understanding of the at once infinite is explained. At the same time the assumed exact nature (and neutrality) of mathematics is questioned (in the spirit of „Koers? as a Christian academic journal). This contemplation disregards the question of what mathematics is (for example by including topology, category theory and topos theory), which would have diverted our attention to contemporary views of figures such as Tait, Penelope and Shapiro who, among others, acts as the editors of and contributors to the encompassing work „Handbook of Philosophy of Mathematics and Logic? (2005).
Highlights
Hierdie artikel handel oor die ontstaan van die moderne versamelingsleer wat, volgens Meschkowski, saamval met die eerste bewys wat Cantor in 1874 gepubliseer het vir die ooraftel
Verskeie wiskundiges en wiskundige tradisies het gedurende die twintigste eeu die gebruik van die opeens-oneindige in die wiskunde bevraagteken
Hierdie besinning sien egter daarvan af om nader op die aard van die wiskunde in te gaan – wat ons gedagtegang sou heenvoer na die kontemporêre opvattings van persone soos Tait, Penelope en Shapiro, onder meer in hulle rol as redakteurs van en bydraers tot die omvangryke “Handbook of Philosophy of Mathematics and Logic” (2005)
Summary
Om te kan tel is sekerlik een van die mees basiese vaardighede waaroor die mens beskik. Die klein kindjie kan onmiddellik sien of daar een, twee of drie voorwerpe voorhande is. ‟n Mens verwys dikwels na die moontlikheid om dit wat teenwoordig of voorhande is oombliklik te sien deur gebruik te maak van die uitdrukking met een oogopslag. Slegs indien alle voorwerpe van mekaar onderskei kan word, word dit moontlik om die hoeveelheid daarvan vas te stel. Daar kan gevolglik ook van twee uitdrukkings gebruik gemaak word wat basies dieselfde sê: onderskeie hoeveelheid en diskrete kwantiteit. Daar is egter ‟n subtiele verskil tussen onderskeie hoeveelheid en diskrete kwantiteit indien die woord onderskeie verwys na dít wat deur die mens onderskei is. Hierdie telwoorde (numerals) moet onderskei word van die gegewe (“nog-nie-getelde”) “aantalligheid” (diskrete kwantiteit) van dít wat getel (kan) word; met ander woorde dit wat telbaar is
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