while (a=b()) ... infinite sets digression

[Michael P. Reilly]

Gordon McMillan <gmcm at hypernet.com> wrote:
: Greg Ewing wrote:

:> Chad Netzer wrote:
:> > 
:> >  So, there are infinitely more strings which start with underscores than do not,
:> 
:> Hmmm... this would seem to imply that there are
:> infinitely many more strings starting with any
:> given character than any other character.
:> 
:> Which seems absurd. Although where infinities
:> are involved, that doesn't necessarily mean it's
:> not true...
:> 
:> Tricky and dangerous beasts, these infinities!

: Unfortunately, Chad mispoke. For any string beginning with an 
: underscrore there are (num_possible_characters-1) strings not 
: beginning with an underscore (just replace that first character).

: But if there are Aleph-nought strings beginning with underscore, 
: there are N*Aleph-nought == Aleph-nought not beginning with 
: underscore.

: Chad's trickery is in trying to get you to assume that Aleph-nough 
: minus Aleph-nought is 0. Nice try.

Next he'll try to convince us that aleph-null == C, right?

: All this assumes, of course, that you allow strings of infinte 
: length. Personally, I don't. And since strings don't have sides, you 
: can't even fake one by folding a finite-lengthed one into a Mobius 
: string.

Turing allowed them. ;)  But you can also create a universe with a
metric defined such that a string of any number of characters still
yields a length of infinity.

  class IString:
    def __init__(self, chars):
      self.value = chars
    def __len__(self):
      # problem is defining aleph_null as a PyIntObject
      return aleph_null
    def __getitem__(self, index):
      """Repeat the contents, ad nauseum, just as an example."""
      return self.value[index%len(self.value)]

  -Arcege