On 3/21/2013 8:29 AM, WM wrote: > On 21 Mrz., 14:02, William Hughes <wpihug...@gmail.com> wrote: > >>> But you think that after all finite and unnecessary lines another one >>> is lurking like a dragon? >> >> Now I think that after any finite set of unnecessary lines has >> been removed, there still remains an unnecessary line.- > > I know. That's what I wished to prove. In order to believe in the > existence of actually infinite sets, it is necessary to have another > element after all ordinary elements have been removed. The set is more > than its elements, namely at least one element more. The actually > infinite set of finite natural numbers remains after all finite > numbers have been removed, since it contains an infinite number. But > that is unmathematical.
That the set is different from its elements is precisely why logical construction by classes takes the form of a class hierarchy.
This is why the rational numbers used to construct Dedekind cuts are not the same as the rational numbers represented by Dedekind cuts.
Because you do not bother to understand these things, you confuse yourself and blame it on Cantor.
But, Cantor explained himself clearly on these matters. Thus the fault lies with you.