On 13/07/2013 09:16 PM, Warp wrote:
> Orchid Win7 v1<voi### [at] devnull>  wrote:
>> I was under the impression that if you allow infinitely long decimal
>> expansions then all reals are representable.
>
> The question would thus be: Can the set of real numbers be represented
> with digits chosen from a finite set?
>
> Or even: Does an infinite amount of digits (chosen from a finite set)
> become uncountably infinite? Doesn't that go contrary to the notion of
> countable sets?

No.

Consider for a moment the set of all natural numbers, N. This is (by 
definition) a countable set. Yet the set of all possible subsets of N is 
*uncountable*. And that's just plain ordinary subsets; the decimal 
expansion of a number is not merely a subset of N but an *ordered 
sequence* of digits. Intuitively it sounds like there should be *more* 
of these. (But, as with many things in set theory, it turns out actually 
the cardinalities are the same.)

Wikipedia explicitly mentions this:

http://en.wikipedia.org/wiki/Cardinality_of_the_continuum#Alternative_explanation_for

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