On 2/25/2012 9:49 PM, Devin Jeanpierre wrote:
What this boils down to is to say that, basically by definition, the set of numbers representable in some finite number of binary digits is countable (just count up in binary value). But the whole of the real numbers are uncountable. The hard part is then accepting that some countable thing is 0% of an uncountable superset. I don't really know of any "proof" of that latter thing, it's something I've accepted axiomatically and then worked out backwards from there.
Informally, if the infinity of counts were some non-zero fraction f of the reals, then there would, in some sense, be 1/f times a many reals as counts, so the count could be expanded to count 1/f reals for each real counted before, and the reals would be countable. But Cantor showed that the reals are not countable.
But as you said, this is all irrelevant for computing. Since the number of finite strings is practically finite, so is the number of algorithms. And even a countable number of algorithms would be a fraction 0, for instance, of the uncountable predicate functions on 0, 1, 2, ... . So we do what we actually can that is of interest.
-- Terry Jan Reedy -- http://mail.python.org/mailman/listinfo/python-list
