Re: Cantor Confusion


*** T. Winter schrieb:

In article <1162299524.423928.41670@xxxxxxxxxxxxxxxxxxxxxxxxxxxx> mueckenh@xxxxxxxxxxxxxxxxx writes:
> *** T. Winter schrieb:
>
> > > The definition is sufficiently special (=3D non-general) to determine
> > > that lim [n-->oo] {1,2,3,...,n} =3D N is correct.
> >
> > Yes, of course, because that was the definition you provided. So because
> > it is so defined, the definition is correct.
> >
> > >
> > > The operator "lim [n-->oo]" defines N. In your example lim {n -> oo}
> > > {-1, 0, 1, ..., n} we have N too but in addition the numbers -1 and 0.
> >
> > "lim {n -> oo}" is an operator that works on sequences, apparently. And
> > you have defined that operator for precisely one sequence of sets.
>
> It is defined for infinitely many sets of integers.

(Infinitely many sequences of sets.)

> lim [n-->oo] {-k,-k+1,..., 0, 1,2,3,...,n} = {-k,-k+1, ...,0} u N for
> every k e N.
> lim [n-->oo] {k, k+1, k+2,...,n} = N \ {1,2,3,...,k-1} for every k e N.

Finally you give a definition. Why did it take so long?

I thought that this was so clear that no explanation was required.

> > > > > case, we have 1/n < epsilon for every positive epsilon and we may
> > > > > *define or put*
> > > > > lim [n-->oo] 1/n = 0.
> > > > > In the second case we have without further ado
> > > > > lim [n-->oo] 1/n = 0.
> > > > > That is the difference between potential and actual infinity.
> > > >
> > > > Well, in mathematics the first form is valid, the second is not
> > > > valid
> > >
> > > If actual infinity is assumed to exist, then the second case is valid.
> >
> > No. The second is never valid, because the limit notation is defined
> > in such a way that the limit point is *never* reached (all definitions
> > of limits in mathematics are formulated in such a way that the limit
> > point itself, or possible function values at the limit point, are not
> > used). And in mathematics 1/oo is *not* defined.
>
> Not in mathematics. But in a theory which assumes omega to be a whole
> number.

Which theory?

Set theory.
Cantor invented omega and defined omega as a whole number.
Who changed this standard meaning?
Why do you think this meaning was changed?
When do you think the contrary meaning became standard?
What is the contrary meaning?
Do you agree that A n: n < omega is incorrect?
If not, why do you complain about on-standard meaning on Cantor's
definition?


> Die Anzahl einer unendlichen Menge [ist] eine durch das Gesetz der
> Zählung mitbestimmte unendliche ganze Zahl. (G. Cantor, Collected
> Works p. 174)
> ... kann also omega sowohl als eine gerade, wie als eine ungerade Zahl
> aufgefaßt werden. (G. Cantor, Collected Works p. 178)

Where in the above quote is 1/oo defined?

It is defined that omega (which Cantor used later instead of oo) is a
number larger than any natural number n. Omega is the limit ordinal
number. Therefore 1/omega must be a number smaller than every fraction
1/n.


> > > > But now you are talking in analysis where limits are properly defined
> > > > (because there is a topology and a metric).
> > >
> > > Set theory was discovered and defined as being based on analysis. Cp.
> > > Cantor's first proof.
> >
> > Yes, so what? Set theory is more basic than analysis. In set theory
> > limits are in general not defined.
>
> Therefore we have there limit ordinal numbers?

Yes. So what? Limits are in general not defined.

In particular the limit of all segments of N is not defined. But set
theory does it.

Especially limits of
sequences of sets are in general not defined.

> > > So apply this knowledge to the case of the balls and the vase.
> >
> > it can not be applied to it.
>
> What a lucky accident!
> Don't you see that the whole aim of Newspeak is to narrow the range of
> thought? In the end we shall make thoughtcrime literally impossible,
> because there will be no words in which to express it. (George Orwell
> in "1984")

Nonsense. Suppose I define an ordering relation on sets. How can I apply
that knowledge to numbers? Defining something in one context does not
make it immediately applicable in another context.

Everything is a set (in ZF). Numbers are sets too.

Regards, WM

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