Continued from last message...
    The set of all integers Z is countable:  I think we can agree that
 Z is infinite.  To show it is countable, count each integer by starting
 the count at zero and work your way out systematically from positive 1
 to negative 1, then positive 2, then negative 2, etc., like so:
    integers ....  -5  -4  -3  -2  -1  0  1  2  3  4  5 ...
   the count       11   9   7   5   3  1  2  4  6  8  10
    It should be clear that no integer is counted twice (and so the
 function is OTO) and every integer is eventually counted (so the
 function is ONTO).  Therefore this defines a function from Z+ to Z that
 has a one-to-one correspondence.  Even though in one sense there seem
 to be more integers than positive integers, the elements of the two
 sets can be paired up one for one.  By the definition of cardinality,
 Z+ has the same cardinality as Z, hence Z is countably infinite.
    This also shows that Z+ = w and Z = w and that w (the infinite
 set of positive integers) equals what looks like 2 * w (all the
 integers, positive and negative). 
    I can also show that the set of all even integers 2Z is countably
 infinite, which by the definition of cardinality, means that this
 infinite set has the same cardinality as Z+, and by the last proof, the
 same cardinality as Z.  This is one way of saying that, as far as
 cardinality goes, Z = Z+ = 2Z.
    If you want, I'll show that the cardinalities of the set of positive
 rational numbers equals that of the set of *all* rational numbers
 equals that of Z+.  I can show that the set of all real numbers between
 0 and 1 is uncountable, and yet this set has the same cardinality as
 the set of *all* real numbers, which is a way of stating that
 w ^ w = w.
    But not tonight.  Ask if you want to see these proofs and I'll type
 them in  I'd do it now but it's late, and I'm ready to crash.
 BS> This is what I mean, and maybe this is all the proof you'll need.
 BS> In any case, it shows you what countably infinite means and gives a
 BS> visual image of my proof to go with the one I referred you to.
 BS> Positive Integers:  1  2  3  4  5  6  7  8  9  10  11  12  13  ...
 BS> All Integers:  0  1 -1  2 -2  3 -3  4 -4   5  -5   6  -6  ...
 BS> Even Integers:  0  2  4  6  8 10 12 14 16  18  20  22  24  ...
 WE> Basically your proof corresponds x with 2 * x or x/2, so this proof
 WE> would apply to uncountable as well as to countable infinity.
    I'm not sure what you're saying here, but I don't think I agree.  By
 definition, a set is countably infinite if, and only if, it has the
 same cardinality as the set of positive integers Z+.  A set is
 countable if, and only if, it is finite or countable infinite.  A set
 that is not countable is called uncountable.  Therefore, my proof would
 not apply to uncountable sets, except maybe as a counterproof.
 WE> For countable, a diagonal enumeration suffices.  Hence rational
 WE> numbers and integers are equi-numerous. But to show that the plane and
 WE> line are equinumerous is more difficult as there is no continuous
 WE> mapping between the two.
    I'm not following where this is leading or how it relates to whether
 or not w ^ w > w or not.
... Deep peace of the quiet earth to you.
--- PPoint 2.05
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