> Ahh I see.  This is one of many counter intuitive things which happens
> with infinities.  Perhaps the issue here is that there are two different
> ways that we tend to think about sizes of things at play here, and it's
> not always made clear which one we're talking about:
>
> 1) finite vs. infinite
> 2) bounded vs. unbounded
>
> #1 here refers the the number of elements within a set, for instance the
> set {3,7,9} has three elements, the set of integers has infinitely many
> elements, and the set of reals has "even more infinitely" many elements
> in a certain well defined sense.
>
> #2 instead relates to some notion of the "size" of the different
> elements within a set.  This is tricky because depending on what you're
> doing you might care about a different definition of what "size" is. For
> example with positive integers you might care about the number of
> digits, but with real numbers you might care about the magnitude of the
> number with relation to the < and > relations (which is certainly *not*
> the same as the number of digits in it!).  Often you just figure it out
> from context.
>
> So this tells you a little bit about the concepts, but I think that some
> examples are useful in getting the mathematical intuitions.  For instance:
>
> * If you have a set of integers, where the number of digits in each
> integer is bounded, then there are only finitely many integers in that
> set (as you said, at most 10^N).
>
> * If you have a set of integers where the number of digits in each
> integer is unbounded, then there are infinitely many integers in that
> set.  For although each particular integer has only a finite number of
> digits, for any particular number N, you can always "look further" and
> find an integer with more than N digits.

I guess the problem is I'm thinking of each "integer" as a set of 
digits. This set has a finite yet unbounded number of elements. But then 
if you make a set of all possible "integers" you don't get a set with a 
finite yet unbounded number of elements, you get a set with an infinite 
number of elements.

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