It is indeed. Or teaching (at least for unorganized persons like me). I've had students raise their hands halfway through a class to ask why I'm going over material I already covered.
Here's your problem (as I see it anyway). Any probability function must integrate or sum to one (they are designed that way). Technically, that isn't always true (when you get into fuzzy probabilities or other more exotic set theories), but it's true here. Since we are assuming a countable infinity (I'll stick with your term rather than denumerable, as it's both less pretentious and clearer), our probability function is a discrete function rather than continuous. So normally it would be easy enough to determine probability values, particularly if we assume that all elements in our set have equal probablity values.
Unfortuantely, assuming both infinite elements and equal probability creates problems. The probability of selecting any single element is 1/n. However, as n is infinite, it increases without bound. Thus, no matter how high a value I choose for my denominator (1/billion, 1/googol, 1/googolplex) I can always go higher. Even though irrational numbers are not in our set, infinity changes everything assuming equal assigned probabilities. So while it is true that Sum(X from X1 to Xn)=1, the probabibility of any individual X (or element in the set, i.e., gods) is 1/n=1/infinity. Which means that the probability of any X value (and single element) is the limit of f(x)=1/n as n approaches infinity in one direction (there are no -n values). This limit is zero. So the probability of selecting any X value is zero. No matter how large I make n in the fraction 1/n, it will always be less than infinity.
Yes, Assuming that all values have equal probabilities, you are talking about an infinite sequence which sums to 1. An infinite set of discrete values need not have zero probablities for each value. For example, there's the classic 1/2+1/4+1/8+1/16...+1/2^infinity=1. But if the probabilities are equal, then they are all 1/n=1/infinity.
It isn't. It's the summation of an infinite number of discrete values. However, as each probability is 1/n where n is infinity, no matter how small I make this fraction (and fractions constitutes a countable infinity), I can always make it smaller. It still has a limit at 0.
I mentioned the dirac function because it isn't a function in the way the term is usually used. It's the limit of a series of functions. In this case, for each element in our set x, p(x)=1/n where n=infinity.
