Path: ...!3.eu.feeder.erje.net!feeder.erje.net!weretis.net!feeder8.news.weretis.net!eternal-september.org!feeder3.eternal-september.org!news.eternal-september.org!.POSTED!not-for-mail
From: Ben Bacarisse <ben@bsb.me.uk>
Newsgroups: sci.math
Subject: Re: WM and end segments...
Date: Mon, 22 Jul 2024 00:13:38 +0100
Organization: A noiseless patient Spider
Lines: 47
Message-ID: <878qxu72zh.fsf@bsb.me.uk>
References: <v7jrk4$7rnq$2@dont-email.me> <87ed7m7349.fsf@bsb.me.uk>
MIME-Version: 1.0
Content-Type: text/plain
Injection-Date: Mon, 22 Jul 2024 01:13:38 +0200 (CEST)
Injection-Info: dont-email.me; posting-host="2d8b7f9aebd53ab53986b5c5490dbfbf";
	logging-data="313244"; mail-complaints-to="abuse@eternal-september.org";	posting-account="U2FsdGVkX1+PM8E1CtTQBAsLBJ815dq4aG3+sT5Eqns="
User-Agent: Gnus/5.13 (Gnus v5.13)
Cancel-Lock: sha1:jdQ88RbyIjPjVku9eE8cjIhW3NE=
	sha1:3xuWtgdnxcg+WZ/WHlZu/gjeBOc=
X-BSB-Auth: 1.57ab24b1945bde902509.20240722001338BST.878qxu72zh.fsf@bsb.me.uk
Bytes: 2770

Ben Bacarisse <ben@bsb.me.uk> writes:

> "Chris M. Thomasson" <chris.m.thomasson.1@gmail.com> writes:
>
>> For some damn reason when I hear end segments from WM I think of a
>> tree. Take the following infinite 2-ary tree that holds the positive
>> integers:
>> ___________________________________________
>>                   0
>>                  / \
>>                 /   \
>>                /     \
>>               /       \
>>              1         2
>>             / \       / \
>>            /   \     /   \
>>           3     4   5     6
>>          / \   / \ / \   / \
>>       .........................
>> ___________________________________________
>>
>> this goes on and on for infinity... We all can see how this can go for
>> infinity, right WM? Wrt trees there are only leaves in a finite view of
>> it. However, the "infinite view" of the tree has no leafs because it never
>> ends... Fair enough? Or too out there?
>
> That's a can of worms in WMaths.  WM has written 734,342,120 nonsense
> posts about binary trees over the years.  It's one of his favourite
> examples to use to bamboozle his poor students.
>
> The infinite binary tree -- simply a graph with node set N and edge set
> (n, 2n+2) (in your numbering) -- is a particular puzzle for WM because

Correction, there are two such edges of course: (n, 2n+1) and (n, 2n+2).

> the node and edge sets are countable but the path set isn't.
>
> Can you see a proof that the infinite rooted paths can be mapped, one to
> one, with an uncountable subset of R?
>
>> ... The infinite one has no leaves.
>
> If you consider graphs in general, they do not have to be infinite to
> have no leaves.

-- 
Ben.