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Re: Distinguishability of paths of the Infinite Binary tree???
Posted:
Dec 26, 2012 9:09 AM
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On Wednesday, December 26, 2012 4:12:18 AM UTC-8, WM wrote:
> On 26 Dez., 09:43, Zuhair <zaljo...@gmail.com> wrote:
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> > I want to note that I'm not claiming to have paradox in the formal
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> > sense, but there is a kind of extreme counter-intuitiveness involved
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> There is nothing depending on any intuition. The CIBT can be
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> constructed in countably many steps, adding one node in every step and
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> never removing anything. That means thare are not more than countable
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> many different configurations. Therefore not more than countably many
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> different things can be distinguished by nodes. Where is any necessity
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> for "intuition" in this clear mathematical argument?
But there is no "clear mathematical argument" there.
Suppose that there were uncountably infinite "different things"
passing through each node. Are they not divided in half by each
node in your binary tree?
I learned a long time ago that if I cut a piece of cake in two I
got two pieces of cake. Sure if I divide too many times all I
end up with are cake crumbs but that's due to the finite nature
of pieces of cake. There are lots of these "paradoxes".
When it comes to sets, if you divide a finte set in half (if you
can) you get two finite sets. If you divide a countably infinite
set in half you get two countably infinite sets. If you divide an
uncountably infinite set in half you get two uncountably infinite sets.
You can't solve your problem by assuming the set of paths traversing
the CIBT is countably infinite, you must prove that is the case.
As it turns out the set of paths is uncountably infinite just as I
supposed in the beginning and Cantor proved it.
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