Re: abundance of irrationals!)

In article <MPG.1ced4405db67c460989c30@xxxxxxxxxxxxxxxxxxxxxxxxx>,
Tony Orlow (aeo6) <aeo6@xxxxxxxxxxx> wrote:

> Virgil said:

> > > Perhaps you mean a string of such
> > > connections, in which case that is also specified by the string of digits
> > > denoting each number, 0 for left branch and 1 for right.
> >
> > That would be a path. A maximal path is one which starts at the root
> > node and either ends at a terminal node from which no further branches
> > extend, or does not stop at all.
> >
> > In an infinite binary tree (with no finite maximal paths) every maximal
> > path contains an infinite sequence of branches.
> >
> > Each node corresponds to the finite path starting at the root node and
> > ending at the given node, so there are only countably many nodes
> > coreesponding to those countably many finite paths.
> >
> > But there are uncountably many maximal ( and therefore inifinte) paths
> > in an infinite binary tree as defined above.
> >
> Holy cow!! Do you ever listen to yourself, or critically examine what you are
> told?

Much better that TO does.
>
> You have infinite maximal paths, with branches infinitely far from the root
> node, but the nodes they connect are a finite distance from the root?

In an infinite path, each branch and each node is only finitely far from
the root node, just as in the naturals, each n in N is finite.

No two nodes can have infinitely many branches or nodes between them and
still be connected. Every connection between two nodes anywhere in any
tree is by a finite sequence of braches and nodes.

> This is almost enough to make me drink the water, or maybe something
> stronger.

If such truths are too strong for you, avoid mathematics.
.

Relevant Pages

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