what is the significance of 1 divided by the various alephs ($\aleph$), the sets of reals, rationals, and so on. I was thinking about the probability of selecting a random real or integer number. I know zero would conventionally be the answer, but I wanted to try to work with infinitesimally small probabilities and was wondering if there had been any other work done on it.
thanks
So first of all this is really a type error. Cardinals don't let you divide. Perhaps instead you mean "an infinite natural number" but there is no such thing. In particular, there is no uniform distribution on an infinite set (in the sense that a given positive probability is assigned to every point).
Those last two sentences assume that we are working in the standard framework. There are nonstandard frameworks which contain true infinite numbers and infinitesimal numbers. In these frameworks, we still can't put a uniform distribution on the natural numbers. However, in this situation, the uniform distribution on an interval looks more like what you might intuitively expect (that each point has an equal, infinitesimal probability). This isn't quite what happens: instead of dividing $1$ by the number of points, we chop the interval up into a "hyperfinite" number of subintervals of equal length and give each of them equal (infinitesimal) probability. This means that we do not really give $\{ x \}$ a positive probability but rather $[x,x+dx]$ where $dx$ is infinitesimal. Proceeding this way when the interval has infinite length is allowed, but because it depends on which "infinity" we picked to be the number of subintervals, it doesn't really mean anything. Thus in reality what is going on is more similar to the standard situation than it looks (as is always the case, in a certain precise sense).
A loosely related notion in number theory is that of natural density; the natural density of a set $A$ in $\mathbb{N}$ is $\lim_{n \to \infty} \frac{|A \cap \{ 1,\dots,n \}|}{n}$. Thus you take the probability of being in $A$ under a uniform distribution on $\{ 1,\dots,n \}$ and send $n$ to infinity. This is useful in certain situations in number theory (for example in describing the distribution of the primes). But it is not a probability measure because it is not countably additive (the natural density of $\{ n \}$ is $0$, the natural density of $\mathbb{N}$ is $1$, but $\bigcup_n \{ n \}=\mathbb{N}$).