Virgil
Posts:
8,833
Registered:
1/6/11


Re: Matheology � 300
Posted:
Jul 13, 2013 3:35 PM


In article <2fa13865ebcf431ea1fff322889609be@googlegroups.com>, Zeit Geist <tucsondrew@me.com> wrote:
> On Saturday, July 13, 2013 7:40:24 AM UTC7, Julio Di Egidio wrote: > > "Zeit Geist" <tucsondrew@me.com> wrote in message > > > > news:41be4197cc38420fa4edb90e196ddc2b@googlegroups.com... > > > > > On Friday, July 12, 2013 1:41:31 PM UTC7, muec...@rz.fhaugsburg.de > > > > > wrote: > > > > >> On Friday, 12 July 2013 19:13:19 UTC+2, Zeit Geist wrote: > > > > >> > > > > >> > It is rather silly to expect the process that creates each of the > > > > >> > Naturals would produce the set of all Naturals, as that set is, > > > > >> > itself, not a Natural. > > > > >> > > > > >> Each natural belongs to a finite initial segment. None of them > > > > >> requires a number that is larger than every natural number. In > > > > >> fact the contrary. If you do not talk about the set, then there is > > > > >> no reason to talk about alephs. > > > > > > > > > > Yes, but for every Natural there is a larger natural, hence the number > > > > > of Naturals is larger than any Natural. > > > > > > > > Since the number of natural numbers is not itself a natural number, that is > > > > a nonsequitur, despite standardly the conclusive statement is correct: > > > > indeed, a fallacy of relevance. Plus, the standard here is in question, so > > > > one should rather qualify statements as well as objections (not that WM > > ever > > > > does it, of course). > > > > The are numbers that are not Natural Numbers. > The number of Naturals Numbers is a number, > and it greater than any finite number, that is to say, > It is greater than any Natural Number. > > Here, number means Cardinality, of course. > > In most Mathematical circles the standard is ZF(C). > Yes, standard Set Theory is being questioned here. > And most who question it here have not come up with > a good reason to reject. Nor have they come up with > a suitable replacement. > > > > > > Why wouldn't I talk about the set of Naturals? > > > > > > > > That there is no such thing as a _set_ N (i.e. a finiteinductive set, an > > > > "unfinished set") is a thesis of *strict finitism* already: > > > > <http://en.wikipedia.org/wiki/Finitism#Classical_finitism_vs._strict_finitis > > m> > > > > Those ideas in Finitism are assumptions. > Although they may lead to consistent systems, > they are far less powerful than a system that assumes > an infinite set. > > I can count head of cattle or stones with a Strictly Finite system. > However, it is very difficult to define a Surface Integral and > most likely impossible to prove FLT in any form of Finitism. > > > > > Julio > > ZG
It is well known that finitist analogues of theorems in standard analysis, when they exist, are much more difficult to prove that the theorems themselves, so nothing is gained except greater difficulty in prooving things. 

