: Reply to above: it's not quite valid to estimate the total number of primes by estimating their "proportion" in the set of intergers, and by furthermore seeing what that fraction tends to as the total set tends to infinity. A similar process would lead to the erroneous conclusion that the set of integers is twice as large as the set of even integers, when in fact their cardinality is the same. Or, for that matter, that the set of integers smaller than 10 (or smaller than a googol, or whatever) has a cardinality of zero. These sorts of seeming paradoxes aren't the result of "almost surely" anyway.