Re: how to list all of the real numbers

Everyone needs to calm down here. If you read Calvin's original post, his real numbers are represented by *paths* in a binary tree. He acknowledges that there are uncountably many such paths:

Thus each endless path corresponds to a real number, and every
real number is represented by a path. Some of the reals
are represented by more than one path [...]

As is well known, the number of paths is uncountable,
though the number of nodes is countable.

The word "list" might be misleading, since that usally implies an enumeration by the natural numbers.

Calvin also notes that one can get any real number to finite accuracy in a finite number steps:

Thus, amazingly enough, we can list ALL of these reals
at once, to any desired binary place, simply by working
on the tree, from left to right, row below row of nodes,
for a finite number of steps. And we can calculate that
number of steps.

'All at once' is misleading, of course, since multiple
reals occupy the same paths to whatever desired binary
place we choose to stop. But they're all there.


I do not share Calvin's amazement, however. To many of us here, the method he proposes is nothing new.

I have the impression that Calvin is earnestly learning about uncountablity and such, and he is sharing his ways of thinking about the issue. He does not strike me as a crank who, for example, insists that the reals must be countable.

The topic of uncountability is so often brought up by cranks that it is very easy to miss the sincerity of non-cranks. (Is there a better word than that?) I think we ought to be careful not to discourage those who come to learn.

--
Stephen J. Herschkorn sjherschko@xxxxxxxxxxxx
Math Tutor on the Internet and in Central New Jersey and Manhattan


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