Re: Cantor Confusion

On 24 Feb., 03:06, "Dik T. Winter" <Dik.Win...@xxxxxx> wrote:
In article <1172251940.593004.240...@xxxxxxxxxxxxxxxxxxxxxxxxxxx> mueck...@xxxxxxxxxxxxxxxxx writes:

> On 23 Feb., 05:31, "Dik T. Winter" <Dik.Win...@xxxxxx> wrote:
> > > On 21 Feb., 16:10, "Dik T. Winter" <Dik.Win...@xxxxxx> wrote:
> > ...
> > > > > like infinity. But it is impossible that both follows, infinity
> > > > > and naturality. The definition a = z for an infinite chain of
> > > > > inequalities a < b < c < ... < z is a *wrong definition*.
> > > >
> > > > What are you *talking* about?
> > >
> > > Simply about 2n - n < 2(n+1) - (n+1).
> >
> > I do still not understand. What are you *talking* about?
>
> The difference 2n - |{2,4,6,...,2n}| steadily inceases with incresing
> n. Only after infinitely many increasings it drops to -aleph_0.

Eh? I would think that 2n - |{2,4,6,...,2n}| equals n, so it increases
without bounds. Why do you think it drops suddenly drops?

I do not think that. Those who say that aleph_0 exists being larger
than any natural number claim it.

> > The (only) infinite initial segment differs from each finite initial
> > segment by having an element that is not in the finite initial segment.
>
> How does it differ from all finite segments? Or is here an occasion
> where the simultaneous consideration of all (segments of) natural
> numbers is inappropriate (quite contrary to Cantor's diagonal proof)?

There is no single number where it differs from *all* finite segments.

So there is no set theoretic indication that it exists as a set other
than a finite set?

But you can do your comparison in parallel, but the comparisons do *not*
give a single number. Why you think that should be the case escapes me.

A set S which differs from all sets A, B, C, ... of a set of sets,
either can do so by by differing from set A by at least one element a
and from set B by at least one element b and so on. Here set A may
contain b and set B may contain a. But in a linear order like A c B c
C c ... this is not possible. Here S must differ by an element from
all sets or it does not differ from all sets.

> > > > Wrong, there is no induction involved.
> > >
> > > Either induction or belief.
> >
> > I think you are on "induction and belief". What induction is there in
> > f(n) = 2 * n? Moreso, what induction is there in (when considering the
> > reals) f(x) = x^3?
>
> You need to find the position of such numbers as [pi*10^10^100] in the
> sequence of natural numbers, but without always having such an easily
> computable short hand as [pi*10^10^100]. There are at least
> [pi*10^10^90] natural numbers which cannot be determined other than by
> counting along the sequence of natural numbers. Therefore inductive
> counting is required. For reals the same is valid, further you have to
> count the digit positions behind the point.

This makes not much sense mathematically. At least I can not determine
any sensible meaning here.

You should turn your interest more to finite numbers than to infinte
sets.

Regards, WM

.

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