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Re: Distinguishability of paths of the Infinite Binary tree???
Posted:
Dec 26, 2012 7:24 AM
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On 26/12/2012 7:29 AM, Zuhair wrote:
> On Dec 26, 12:21 pm, Virgil <vir...@ligriv.com> wrote:
>> In article
>> <58fbc0c5-854b-4c63-bd5f-58faa3908...@d4g2000vbw.googlegroups.com>,
>>
>> Zuhair <zaljo...@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.
>>
> Yes but a countable set of them suffices! no?
Of course. And how many such countable sets are there? Cantor showed
that there are uncountably many.
There are at least two counter-intuitive notions when dealing with
infinities: There is an infinite set, each of whose elements are finite
(viz. the sequence of initial segments {{1}, {1,2}, {1,2,3}, ...}; and
the set of all countable subsets of a countable set is uncountable. The
only thing this shows is that intuition is sometimes insufficient to
grasp complex things.
>>> 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.
>
> Possibly I don't know, but there is some Intuitive issue that WM is
> addressing. Anyhow those kinds of discussion are not really easy to
> run because they are discussions at Truth level which is in a sense
> higher than just formal level. One can always still keep insisting
> that all sets are countable and that uncountability is just a form of
> a Pseudo-argument as far as reality of the matters is concerned like
> saying that the quantifiers in Cantor's argument can only be suitably
> understood to be first order, i.e. ranging over "elements" of the
> universe of discourse, and so doesn't cover ALL functions in reality,
> because some functions (which are subsets of the universe of
> discourse) might not be elements of the universe of discourse! that is
> usually the basis for it being possible to have a countable model of a
> theory that proves existence of uncountably many objects, and in this
> scenario uncountability pops up as an artifact due to a defect in the
> theory's ability to define all functions and not due to something that
> reflects some issue that is present in the real world of sets. Those
> kinds of arguments might really be motivated by presenting strong
> intuitive similes against Cantor like that one present here, albeit
> I'm not so sure if the point that I've presented here is sufficient
> for such a drastic alternative move. Anyhow that doesn't mean that
> Uncountability is not interesting, in reality it is, even if it is
> just a kind of internal (intra-theory) manifestation, but by then it
> would only be interesting at formal level. It won't have any
> philosophical significance. Anyhow.
>
>
>> --
>
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