Prime numbers are a well-defined set with specific membership criteria. Can the same be said about "numbers"? Aren't numbers (that is, all numbers) a well defined set but without membership criteria? Anybody can say, given a particular object, whether that belongs in the set of numbers or not. But it may not be possible to give any criteria for this inclusion.
We may want to say that the set of all numbers has a criterion and that is that only numbers shall get into the set and all non-numbers shall stay out of it. But then, this is merely a definition and not a criterion for inclusion.
Therefore my question: Can there be a well-defined set with no membership criteria?