On 19 Jun., 22:17, Zeit Geist <tucsond...@me.com> wrote: > On Wednesday, June 19, 2013 11:38:03 AM UTC-7, WM wrote: > > On 19 Jun., 19:04, Zeit Geist <tucsond...@me.com> wrote: > > > > Anyway, what Cantor wrote is NOT the theory of ZFC. > > > In modern ZFC omega is used as ordinal and as cardinal simultaneously. > > As if you know.
I have frequently been taught so by modern set theorists. I assume they are knowledgeable with respect to their business.
> Every step in the sequence gives a finite set which is NOT all N.
Correct. But the whole sequence has infinitely many steps and infinitely many unions.
> If you take the entire set of the sets constructed at each step > ( each of these sets are finite, but there infinitely many of them ),
then there are aleph_0 numbers missing in every set of the sequence, which contains the result of infinitely many unions.
> and then take the Union over that set; then the result is N.
Just that is surprising. Infinitely many + 1 union are sufficient. > > So what if it takes aleph_0 step to form the list, and then one more > step for the Union to arrive at a set containing all and only all N.
Surprisingly infinitely many unions (where each union adds something) are not sufficient. But another union that does not add anything yields |N.
> Start you list at 0, the least ordinal, and this makes more sense.
Why: In this case the same happens: Infinitely many unions, where each union adds something, cannot yield |N. But another union that adds nothing, yields |N. Not mor and not less sense.
> Or don't and remain lost in your blather.
That sound as if you have run out of arguments. > > As for you last bit from Wittgenstein, this only shows > that omega ( infinity ) is inaccessible via succession ( counting ).
Then it is not useful for analysis. There we need lim n --> oo. Like in my infinitely many unions.
> That's why ZF(C) has an Axiom of Infinity.
This axiom has been made by Zermelo, taken from Dedekind. There it meant nothing that can be reached but potential infinity.
> Which does not contradict any logical principle.
It contradicts the principle that unioning a set with its subsets canot increase that set.
> It may contradict our everyday intuition; but if we restricted > our selves to that boundary, we would still be in the > Dark Ages.
Someone used to say: Ich führe euch herrlichen Zeiten entgegen. Now these times have dawned - also in matheology. > > BTW, you still haven't given me the proper translation > that I asked for in the last post.-
I have only the German text. It is from "Georg Cantor: Leben, Werk und Wirkung", by Herbert Meschkowski, 2. Aufl. BI, Mannheim (1981) p.142. It quotes G. Kowalewski, Bestand und Wandel, p. 202.