On-line version ISSN 2304-8557
Print version ISSN 0023-270X
STRAUSS, D.F.M.. (Non)denumerability. Koers (Online) [online]. 2011, vol.76, n.4, pp.637-659. ISSN 2304-8557.
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).
Keywords : (non-)exact; at once infinite; denumerable; diagonal proof; infinite totality; non-denumerable; successive infinite.