On 14 Mrz., 19:52, fom <fomJ...@nyms.net> wrote: > On 3/14/2013 5:52 AM, WM wrote:
> >> A particular claim must be made before Cantor's argument > >> can even apply. > > >> That particular claim is that there is a single infinity > >> that is referred to in language as an object. > > > No. That particular claim is that infinity can be complete > > Wrong. The completeness issue is entirely separate > from the diagonal proof.
Cantor's opinion about his proof is wrong? > > It is clear that you completely fail to understand > that.
Cantor failed to understand his proof too? Of course you may be better than he, or you may at least think so.
> Whether this is because your general feelings > concerning infinity make you unable to see the difference > or simply because you are pursuing an agenda makes > no difference.
It is because I have read and understood Cantor's arguments. > > In the Grundlagen (or, at least in the translations which > my ignorance forces me to use), Cantor explains in > detail why the logical construction of real numbers > from sets of rationals is reasonable and how one should > think about accepting the construction as legitimate.
Then you should also read his thoughts about infinity. > > Nor can one attribute the completeness issue to Cantor > alone since Dedekind was addressing the same issue > differently.
Dedekind got his idea of infinity from Bolzano. I think that I think that I think ... That is never finished. Zermelo got his idea of infinity (his axiom) from Dedekind, but inadvertently changed its meaning to actuality, because that is required for Cantor: There is a set, that means there are all elements of the set. > > You would be correct to say that Cantor believed in > a completed infinity so that the diagonal argument > motivated his further researches. But, you need > to differentiate what the proof does and how it > does it from other mathematics that involves > completed infinities
It is so simple: If the list is only potentially infinite, then it is not possible to decide whether a number can be excluded definitively. But it is necessary for Cantor's argument to exclude the diagonal from the whole list - not only from the first n lines for every n, because after every n there follow infinitely many more. It is not possible to judge about the complete list because incomplete infinity is incomplete. Is that so difficult to understand?
Here it is again. Think about it: It is not possible in potential infinity to judge about the complete list because incomplete infinity is incomplete.