Re: Cantor Confusion

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?

> > > 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?

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?

> > 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. 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.

--
*** t. winter, cwi, kruislaan 413, 1098 sj amsterdam, nederland, +31205924131
home: bovenover 215, 1025 jn amsterdam, nederland; http://www.cwi.nl/~***/
.

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