Re: Another set with cardinality |Z|

From: Robert Low (mtx014_at_linux.services.coventry.ac.uk)
Date: 09/26/04

  • Next message: Owen Jacobson: "Re: Another set with cardinality |Z|" 
    Date: 26 Sep 2004 08:42:54 GMT

    Eray Ozkural exa <erayo@bilkent.edu.tr> wrote:
    >mtx014@linux.services.coventry.ac.uk (Robert Low) wrote in message
    >> but they are still all finite. In the same way,
    >> there is no upper limit to the depth of branches
    >> in the tree, but they are still all finite.
    >
    >Excuse me, I am not interested in the vertex labels or depths of paths
    >from a chosen vertex to the root.

    Well, you ought to be, since the length of a path from the root to
    a leaf is the number of bits in the binary expansion of the number
    that path represents. And they are *all* finite.

    >I am asking the depth of the limiting tree. A finite number d, or
    >countably infinite as you seem to suggest above? I guess we both want
    >to say the latter.

    The limiting tree has infinite depth in the sense that
    there is no upper limit to the leaf-root distance. But
    there is no, I repeat, *no* path in this tree that
    corresponds to an infinite binary expansion.

    >Ok. We can work with that. So, how many vertices are there in the
    >limiting tree? |Z| as we expected?

    Of course. It's the union of a countable number of finite
    sets, so it must be countable. 

    -- 
Rob.  http://www.mis.coventry.ac.uk/~mtx014/
  • Next message: Owen Jacobson: "Re: Another set with cardinality |Z|"

    Relevant Pages

    • Re: Another set with cardinality |Z|
      ... a leaf is the number of bits in the binary expansion of the number ... >countably infinite as you seem to suggest above? ... The limiting tree has infinite depth in the sense that ... there is no upper limit to the leaf-root distance. ...
      (sci.math)
    • Re: Another set with cardinality |Z|
      ... > exactly like in the infinite depth. ... I took the limiting tree as in Robert Low's post, ... concatenates all the digits from the root. ... infinite bit strings or not? ...
      (comp.theory)
    • Re: Another set with cardinality |Z|
      ... > exactly like in the infinite depth. ... I took the limiting tree as in Robert Low's post, ... concatenates all the digits from the root. ... infinite bit strings or not? ...
      (sci.math)
    • Re: infinity
      ... >>> applying the definition of what an infinite sum means. ... > This is a standard application of the definition of an infinite series. ... > But n is never at its upper limit. ... That's why we write the infinite sum in that way. ...
      (sci.math)
    • Re: How many horses?
      ... >> My solution was very much in the spirit of chess. ... No you weren't.......The answer that 32 is an upper limit is not ... integer numbers are infinite since integers are infinite, ...
      (rec.games.chess.computer)