Virgil
Posts:
4,482
Registered:
1/6/11
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Re: Distinguishability of paths of the Infinite Binary tree???
Posted:
Dec 26, 2012 4:21 AM
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In article
<58fbc0c5-854b-4c63-bd5f-58faa3908be7@d4g2000vbw.googlegroups.com>,
Zuhair <zaljohar@gmail.com> wrote:
> On Dec 24, 12:42 pm, WM <mueck...@rz.fh-augsburg.de> wrote:
> >
> > There is nothing to happen "in the infinite". And it is completely
> > irrelevant whether the paths after the distinction are finite or
> > infinite. Everything that happens in a Cantor-list and in a Binary
> > Tree happens at a finite level.
> >
>
> Up till now nobody have answered my question, anyhow. I still find it
> puzzling really, Cantor has formally proved that there are more
> distinguishable reals than are distinguishable finite initial segments
> of them, I find that strange since the reals are only distinguishable
> by those initial segments, so how they can be more than what makes
> them distinguishable? This is too counter-intuitive!?
Note, however, that there is no finite initial segment of any one
infinite binary sequence that distinguishes it from ALL others.
>
> Probably this counter-intuitive issue is similar to the conflict
> between distinguishability and the number of elements of a proper
> subset and its set at infinite level, where the set would have
> strictly more distinguishable elements than a proper subset of it and
> yet they both have the SAME number of elements. So it appears to me
> that the number of elements of infinite sets departs from the notion
> of distinguishability.
Depends on the level of distinguishability at issue.
For any finite set of such strings, finite initial segments suffice to
distinguish all of them from each oterhbut for at least some infinite
set, no finite set of finite initial segments suffices.
>
> I want to note that I'm not claiming to have paradox in the formal
> sense, but there is a kind of extreme counter-intuitiveness involved
> here with the notion of uncountability. Indeed this might drive some
> to reject being involved with such concepts that would mess about our
> intuitive faculaties and they would maintain that such slippery areas
> of ideation are better avoided than engaged since they might be too
> misleading. Anyhow
What drives WM is shear orneryness.
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