1887

n Koers : Bulletin for Christian Scholarship = Koers : Bulletin vir Christelike Wetenskap - (Oor)aftelbaarheid

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Abstract

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 ooraftelbaarheid van die reële getalle. Later het Cantor sy bekende diagonaalbewys ontwikkel, wat die kern van hierdie artikel vorm. Die beredenering van die onderhawige artikel is gerig op die implisiete veronderstelling van die diagonaalbewys, naamlik die aanvaarding van die aktueel-oneindige (wat meer gepas die opeens-oneindige genoem kan word). Sonder hierdie aanname kan nie tot ooraftelbaarheid gekonkludeer word nie. Verskeie wiskundiges en wiskundige tradisies het gedurende die twintigste eeu die gebruik van die opeens-oneindige in die wiskunde bevraagteken. In die besonder word nader ingegaan op twee prominente teenstanders van die opeens-oneindige, naamlik Kaufmann en Wolff. Die sirkelredenasie wat in albei benaderings opgesluit lê, word uitgewys en as alternatief word 'n verantwoording van die gebruik van die opeens-oneindige verduidelik wat nie dit wat bewys wil word as uitgangspunt neem nie. Tegelyk word die beweerde eksaktheid (en neutraliteit) van die wiskunde bevraagteken (in die gees van "Koers" as Christelikwetenskaplike tydskrif). Hierdie besinning sien egter daarvan af om nader op die aard van die wiskunde in te gaan (deur byvoorbeeld ook die topologie, kategorieteorie en toposteorie te betrek) - 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).


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).

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/content/koers/76/4/EJC126374
2011-01-01
2016-12-09
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