I hope nobody minds my resurrecting this thread. I wanted to mention that if we work in ZFC, the standard axiom system for modern math, there are sets that satisfy neither of those properties.
Consider the set of real numbers; and if x and y are real numbers, define the relation x ~ y if it happens to be the case that x - y is a rational number. For example, (1/3 + pi) ~ pi because (1/3 + pi) - pi = 1/3 is rational.
This is readily seen to be an equivalence relation. For all real numbers x, we have x ~ x; also for all x and y we have x ~ y implies y ~ x; and for x, y, and z we have x ~ y and y ~ z implies x ~ z.
This equivalence relation partitions the real numbers into a set of mutually disjoint equivalence classes. By the Axiom of Choice (that's the C in ZFC) there is a set, call it V, that consists of exactly one representative from each of the equivalence classes.
We know from the rules of set theory that V exists. We can not name any of its specific members; nor is there any property, other than membership in V, that's shared by the elements of V. We can't look at a real number and say whether that number is in V. If you ask, "Is 1/3 in V?" I answer that I don't know. I do know that V contains exactly one rational (why?) but I have no idea which one it is.
V is a set whose members are not known, whose members can not be recognized as belonging or not belonging to V, and share no property in common.
I should mention that this kind of construction is exactly why some people don't like the Axiom of Choice. However if you reject Choice, you get other unpleasant anomolies. So mathematicians just accept it and get used to using it. What that means metaphysically is a personal choice.