Re: Cantor Confusion

In article <1161518008.776999.238550@xxxxxxxxxxxxxxxxxxxxxxxxxxxx> mueckenh@xxxxxxxxxxxxxxxxx writes:
*** T. Winter schrieb:
In article <1161435575.019298.164830@xxxxxxxxxxxxxxxxxxxxxxxxxxx> mueckenh@xxxxxxxxxxxxxxxxx writes:
....
> > How do you *define* X(omega). As far as I know X is only defined
> > for real numbers, and omega is not one of them. And I see no reason
> > to exclude X(omega) = 0, = 1, = -1 at all from this reasoning.
>
> lim {t --> oo} X(t) = X(omega)

Yes, if you define X(omega) like that your conclusion is obvious.

Not I define omega as the limit ordinal number. That is a matter of set
theory.

You use oo, which is *not* the same as omega. But when we ignore that
difference, more problems prop up. X(t) was a function from R to R, I
think, not a function from ordinals to something else. But you stated
that X(t) was 9.t. Again lacking precision. On the one hand,
if X(t) = 9.t, we get:
lim{t -> omega} X(t) = X(omega)
but with X(t) = t.9, we get:
lim{t -> omega} X(t) != X(omega).
Yes, in the ordinals multiplication is not commutative...

Not I define omega as the limit ordinal number. That is a matter of set
theory. I only use the continuity of a function which is already
required to find lim 1/n = 0.

There is no continuity required to find such a limit. Limits are independent
on continuity. The function:
f(x) = 1 when x = 0 and = 0 when x != 0
is certainly not continuous at x = 0, nevertheless the limit for x -> 0
exists, and is 0. What continuity do you use when you calculate:
lim{x -> 0} f(x)
?

> lim {t --> omega} t = omega

Oh. Provide a mathematical definition of that limit, please. In standard
mathematics that limit is undefined.

Cantor used omega with two slightly different meanings. omega is the
set N and omega is the first infinite ordinal number, i.e., the
smallest number larger than any natural. These two definitions yield:
lim {t --> oo} t = omega
and
lim {t --> oo} {1,2,3,...,t} = omega.

Yes, so what? I asked for a mathematical definition, not for handwaving.
And none of the usages of Cantor do in any way define the limit. What
*is* your mathematical definition of that limit?

> > You think so. The irrational numbers are defined to be the limits
> > of some particular sequences (or rather as equivalence classes of
> > sequences). I
>
> Equivalence classes of sequences with same limit like
> lim {t --> oo} a_t.

Wrong.

Wrong is wrong. The limit *is* the irrational number. You can use
these and only these numbers in a Cantor list, not the equivalence
classes of sequences.

You really do not understand how the reals are defined. The limit is
*not* the irrational number. The limit does not even exist.


> > (7) assume sequences of rationals. Create equivalence classes amongst
> > those sequences (a_n ~ b_n if |a_n - b_n| goes to 0; but this is
> > losely speaking and quite a few other methods are known, all
> > equivalent).
...
> > So, at what stage in this process is the limit of a function used to
> > define the irrationals?
>
> At (7). The equivalence classes of sequences of rationals with same
> limit.

Wrong. At that point you can not talk about sequences of rationals with
the same limit, because many of such sequences do not have a limit in the
rationals. So (7) is formulated as I wrote it (in one of the forms to
define the reals from the rationals). It is not the *limit* that is the
irrational, it is the equivalence class of sequences.

The sequences belong to Q. So your irrational numbers belong to Q? That
is nonsense. In Q we have sequences with Cauchy-convergence and,
therefore, perhaps without a limit in Q. But the irrational numbers are
definitely *not* in Q.

Pray re-read what I wrote. The real numbers are defined as equivalence
classes of sequences of rational numbers. The sequences do not belong to
Q. They are sequences of elements of Q. So when defining reals (according
to this methodology) we start with sequences of rationals. We call two
sequences equivalent if their difference goes to 0, the concept of limit
has not yet even been defined. It is easily shown that that is an
equivalence relation, so we can divide the sequences in equivalence classes.
Each of those equivalence classes is a real number. So a real number is
an equivalence class of rationals. It is easy to embed the rational
numbers isomorphically in the set of real numbers (when arithmetic on the
real numbers is properly defined).

And (again,
losely speaking) the equivalence classes are built in such a way that
all members of the classe *ought* to have the same limit in the extended
system.

Exactly. And that is the irrational number, *not* loosely speaking.

It is *extremely* losely speaking. The real (not irrational) number is
an equivalence class, it is not a limit. So initially, 1/2 is *not*
an element of this new system. An element of this new system is an
equivalence class of which the sequence:
1/2, 1/2, 1/2, ...
is a representative.

You
can use this and only this number in a Cantor list, not the equivalence
class of sequences (because the due terms are not uniquely defined).

Oh, perhaps, I do not understand at all. I do not see the relation.

Summarizing the original question: You need the limit omega to
construct the irrational numbers.

Where in the construction above did I use the limit omega?
--
*** 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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