Search All of the Math Forum:
Views expressed in these public forums are not endorsed by
NCTM or The Math Forum.



Cantor and the axioms (2)
Posted:
Feb 22, 2014 7:46 AM


Aren't a set and its corresponding cardinal number quite different things? Does not the first face us as an object whereas the latter is an abstract picture of it in our mind? {{Remember the roots: A set is an object (of reality).}} The old, so often repeated sentence: "The whole is more than its part" may be accepted, without proof, only with respect to the entities which the whoe and the part are based upon. Then and only then the sentence is an immediate concequence of the notions "whole" and "part" Unfortunately, however, this "axiom" has been used uncountably often without any grounds and neglecting the necessary distinction between "reality" and "magnitude" just in that meaning which makes it false in general, as soon a actually infinite sets are involved and which makes it right for finite sets only because we are able to prove it as right in this domain. [E. Zermelo: "Georg Cantor, Gesammelte Abhandlungen mathematischen und philosophischen Inhalts", Springer (1932) p. 416f]
Regards, WM



