Def.1 Let p be a property of a set A such that for the union of a set A and a set B (B not equal to A), the set minus operator for set A is undefined; i.e. (A U B)\A is undefined.

i) Then p must also be a property of B as else would be determined the elements of A and B.

ii) If p is a property of B then if B is the absolute complement to A then all members are accounted for. As consequence of Def.1 and given ii) that p is now also a property of B, once A is union to B the sets may no longer again be independent of each other.

iii) So we let B be strictly less than the absolute complement to A, and doing so returns us to i) unless p can exist as two distinct properties.

Conclusion There can exist no such standard set. (Per my definition of "we don't know what any of the elements are".)