Re: Cantor Confusion


*** T. Winter schrieb:

> > > Please give me all the bits of 1/3. Then I will show the bijection.
> >
> > I can't so you can not show a bijection.
>
> Representations (= paths) which do not exist cannot be part of a
> bijection.

So you can not show a bijection (in your opinion), but nevertheless you
state that you have given a surjection. Do you not think you are
contradicting yourself a bit?

I give a surjection on all existing paths.

> No? Even if there are enough parts to gather more than a whole edge to
> be mapped on every path?

No. Each part of an edge may map to a single path, that does *not* give
a map from each edge to a single path.

Since infinity = infinity became an allowed equation, 1/2 + 1/2 = 1 can
no longer be true?

I can. And I see that each edge maps, when you do this, to a
plethora of paths, so that is *not* a surjection from edges to paths.
The crucial thing in a surjection (and indeed for a mapping) from A to B
is that each element of A maps to a *single* element of B. So, let me
ask a different question. To which single path does the edge that goes
left from the root map.

That one which goes left from the root is not engaged, because we need
only half of the set of edges. That one which goes right, could be
mapped on that real number (path) which is asked for most frequently,
namely 1/3. That edge leaving the node 1 on level 1 left is mapped on
the next frequently mentioned number, namely pi. You will not be able
to ask for more nmbers than I can name edges. And you will not be able
to construct a "diagonal" path.

But considering this catalog for another time, I remember that we have
twice as much edges as paths. Therefore, we can waste the first
countable infinity of edges, and if not a whole infinity, we can at
least waste all the edges up to level n (for any finite number n) and
notwithstanding this big loss, there remain enough edges to attach one
to each path. That is so overwhelming, that I in fact cannot
understand how you may dare to defend your deplorably swaying position.

My sympathy and condolences.

Regards, WM

.

Quantcast