
Re: Distinguishability of paths of the Infinite Binary tree???
Posted:
Dec 26, 2012 7:24 AM


On 26/12/2012 7:29 AM, Zuhair wrote: > On Dec 26, 12:21 pm, Virgil <vir...@ligriv.com> wrote: >> In article >> <58fbc0c5854b4c63bd5f58faa3908...@d4g2000vbw.googlegroups.com>, >> >> Zuhair <zaljo...@gmail.com> wrote: >>> On Dec 24, 12:42 pm, WM <mueck...@rz.fhaugsburg.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 Cantorlist 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 counterintuitive!? >> >> Note, however, that there is no finite initial segment of any one >> infinite binary sequence that distinguishes it from ALL others. >> >> >> >>> Probably this counterintuitive 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 counterintuitive 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 counterintuitiveness 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 Pseudoargument 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 (intratheory) manifestation, but by then it > would only be interesting at formal level. It won't have any > philosophical significance. Anyhow. > > >>  >

