[hzhou321]

The discussion really helps me understand the problem. Real numbers are infinity in disguise. Because infinity is not really defined, real numbers are not really defined. Just saying infinity is not finite does not make much sense (in the sense of adding anything helpful). Any real number that you can finitely describe can be included in a finitely described counting scheme. Let's use the counting schemes of rational numbers, I'll hit arbitrary numbers to arbitrary precisions. Taking a limit, it is not clear that I miss any numbers (including pi). To defeat this scheme, the adversary has to keep adding digits to his real number, as well as keep shrinking his allowed precision infinitely. Now both the number (that is being infinitely being described) and my counting process is infinite, we didn't and can't prove anything. There is simply no conclusions to be drawn.