[js6i]

Maybe not the most formal of meanings, but my favorite is a probabilistic one: given a random element, how likely it is that it satisfies a predicate? If some elements don't, but it's still satisfied with probability 1, that's pretty clearly almost always.

EDIT: yeah you guys are right, I wouldn't worry too much about the prior not being a proper distribution, but still - this doesn't seem related to the cardinality of sets in a simple way after all!

[Nevermark]

I think for infinitely sized sets this isn't so simple.

For instance, what is the random distribution you are using to select from a set of infinite numbers?

The cardinality of any two intervals of real numbers is the same, regardless of the lengths of the intervals.

[btilly]

Random according to what distribution?

The probability distribution that we all want to define on the real numbers does not exist. And this has non-trivial consequences.