Re: Cantor and the binary tree

In article <1119026531.459290.53390@xxxxxxxxxxxxxxxxxxxxxxxxxxxx> mueckenh@xxxxxxxxxxxxxxxxx writes:
> *** T. Winter wrote:
> > In article <1118841142.975988.63790@xxxxxxxxxxxxxxxxxxxxxxxxxxxx> muecken=
> h@xxxxxxxxxxxxxxxxx writes:
....
> Numbers which do not differ at any finite bit are not different.

Right.

> Paths which do not differ in the tree by at least one node do not
> differ at all.

Right, and this is the same.

> Bunches of paths which have at least one node of their
> own indicate by this fact that they belong to a countable set.

Wrong. You have to prove that. At each node there emanates an infinite
(and it is uncountable) number of new paths.

> > Yes? So what? For surjectivity there is a requirement that such a set
> > exists. I see you write: "We will see now, that the impossible set does
> > not exist and that the paradox-generating requirement cannot be
> > satisfied, even if the mapping is defined between equivalent set."
> > However, in the example you give there is no such paradox-generating
> > requirement. You apparently still do not get it. In a mapping
> > N -> P(N), if it is surjective, there *must* be an m such that f(m) =3D M.
> > In the mapping {1, a} -> {{}, {1}}, there is *no* such requirement.
> > The requirement here is that either f(1) =3D M or f(a) =3D M. And that
> > requirement is satisfied.
>
> No, the requirement is to map a natural number on the set of
> non-generators with respect to natural numbers.

Do you understand why Hessenberg thinks it is a required mapping? I think
not. The reason is that if you have a map from N to whatever, to be
surjective each element of whatever *must* have a source in N, so there
should be an individual element of N (say m) that maps onto it. For
surjectivity of {1, a} -> {{}, {1}} there is no such need. In that case
the requirement is different.

> This has nothing to do with cardinality, as can be shown by the
> following bijection from N on the union of the set of even numbers and
> the set of nongenerators under the mapping.
> f: N--> E u {non-generators under the actual mapping}.

This is pretty close to nonsensical. How do you define non-generators under
this mapping? I have no idea. As I see on the right-hand side a set of
numbers united with a set of undefined things... Moreover, the right-hand
side depends on the actual mapping, but that is ridiculous.

> You may try to construct any bijection. It will fail, but not because
> there are too less elements in N.

I would not even dare to try a bijection until I know what is meant with
that set on the right-hand side.

> > > Try the well-ordered set of real numbers to construct it by the
> > > diagonal argument.
> >
> > If you well-order the reals it is not a list. Not every well-ordered
> > set is a list.
> >
> But it is defined which element comes first, which comes second and so
> on.

Indeed, but that is not what makes a list.

> Sorry, I have it only present in German. Cantor writes: Unter einer
> wohlgeordneten Menge ist jede wohldefinierte Menge zu verstehen, bei
> welcher die Elemente durch eine bestimmt vorgegebene Sukzession
> miteinander verbunden sind, welcher gem=E4=DF es ein erstes Element der
> Menge gibt und sowohl auf jedes einzelne Element (falls es nicht das
> letzte in der Sukzession ist) ein bestimmtes anderes folgt, wie auch zu
> jeder beliebigen endlichen oder unendlichen Menge von Elementen ein
> bestimmtes Element geh=F6rt, welches das ihnen allen n=E4chstfolgende
> Element in der Sukzession ist ( es sei denn, da=DF es ein ihnen allen in
> der Sukzession folgendes =FCberhaupt nicht gibt).

For the German-impaired: "With a well-ordered set we understand a well-defined
set where the elements are connected through a particular successor function,
where there is a first element of the set and also for each element (if it is
not the last in the set) another element follows, and also such that with
each finite or infinite subset of elements there is a particular element for
which all following elements are part of the successor function" (And I do
not understand the part in parenthesis that follows.)

Indeed. Pretty similar to well-ordering (if I understand it right: each
subset has a first element). But not a list. A list is an order-preserving
mapping from (an initial segment of) the naturals to the elements of the
list. When we order the naturals as:
1, 3, 5, 7, ..., 2, 4, 6, 8, ...
we have a well-ordering, but not a list.

> So, if transfinite numbers are called whole numbers, then a well
> ordering may be called a sequence or a list, although there is an
> element, like omega with respect to N, which is the "next one" to all
> the finite elements n e N. =20

But it is not a list according to the definitions. You may call it a
sequence or a list, but that is not true when you use standard
definitions. That you call those numbers whole numbers does not make
a well-ordering a list. A list is closely tied to the concept of
natural numbers. Could you provide us with the definition Cantor gives
of list?
--
*** 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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