Re: Cantor and the binary tree

Alan Morgan said:
> In article <MPG.1cfe8fc8b6443819989d2c@xxxxxxxxxxxxxxxxxxxxxxxxx>,
> Tony Orlow (aeo6) <aeo6@xxxxxxxxxxx> wrote:
> >Virgil said:
> >> In article <1117020117.383463.212780@xxxxxxxxxxxxxxxxxxxxxxxxxxxx>,
> >> mueckenh@xxxxxxxxxxxxxxxxx wrote:
> >>
> >> > Ron Sperber wrote:
> >> >
> >> > line number n
> >> > 0 0.
> >> > 1 0 1
> >> > 2 0 1 0 1
> >> > ... ..................
> >> >
> >> > > It simply boggles my mind that this simple proof gives so many people
> >> > > such fits that they refuse to accept it. I continue to be sadly shocked
> >> > > by the number of posts on sci.math daily refuting Cantor's proof. Of
> >> > > course they are always fuzzy on details, but that's to be expected since
> >> > > they can't actually disprove it.
> >> >
> >> > You are talking about Cantor's proof. You could say the same about
> >> > mine. I do not say, here, that Canor's proof is wrong. All I say, here,
> >> > is that another proof leads to another result.
> >>
> >> Not if one analyses it correctly.
> >>
> >> Each leaf node (end node of a finite path) corresponds to a terminating
> >> binary proper fraction.
> >>
> >> Each unending path, having no leaf node, corresponds to a
> >> non-terminating binary proper fraction.
> >>
> >> There is no one-to-one correspondence between the set of terminating and
> >> the set of non-teminating, despite WM's weaselings.
> >>
> >That has nothing to do with his point. I'll grant that in an infinite binary
> >tree there are almost surely more infinite paths than finite ones, but that is
> >irrelevant. WM is talking about paths vs nodes and branches, because you are
> >making some bizarre unfounded claim about the paths being uncountable and the
> >nodes countable, when there are half as many paths as nodes.
>
> In a finite tree there are half as many paths as nodes. I totally agree
> with this statement (okay, (nodes+1)/2. Big deal). I think you'll get
> general agreement with this statement.

I should hope so.

>
> In an infinite tree there are as many *finite* paths as nodes. Again,
> I think you can count on general agreement (although there will be
> some grumbling that there are infinite numbers of both and I may be
> off by a factor of 2 but let's forget about that).

If "there are as many *finite* paths as nodes", then what do the infinite paths
consist of? Don't they have nodes as well? Are there infinite paths, if all the
nodes are used up in the finite paths?

>
> I await, with some interest, your proof that, in an infinte tree, there
> are half as many paths (or just as many paths or twice as many paths or
> whatever) *of* *all* *kinds* as there are nodes.

Good. I got called into duty yesterday before I could finish it but it's posted
now. And, no leaf nodes or lack thereof for Virgil to chew on, redundantly
reiterating the same repetitive statement verbatim over and over.

>
> When WM was talking about adding a node being the same as adding a path
> he was, although he didn't know it, talking about *finite* paths. Adding
> a node *is* the same as adding a *finite* path. The situation with
> infinite paths is much messier.
Nope, not. It requires two nodes to make a new path, anyway. One creates a new
path when one adds a second child to a node. refer to my demonstration of node
insertion.
>
> >This kind of
> >result is what brings upon the Cantorians all their woes in the form of us
> >"idiots".
>
> No, the source of the woes is anti-Cantorians insistence that infinite and
> finite are pretty much the same, except that infinite is larger. Anti-
> cantorians think that the integers include infinitely large values.
> Cantorians think that the integers include arbitrarily large values, but
> "know" that arbitrarily large is not infinite. Anti-cantorians think that
> infinite paths are like finite paths, but longer. Cantorians actually
> puzzle out what in the heck is meant by an infinitely long path and
> conclude that it behaves completely differently from finite ones.
>
> Alan
>
Cantorians pretend they know what they are talking about, but always seem to
frget the details of the subject matter along the way.
--
Smiles,

Tony
.

Relevant Pages

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