Is it possible to construct such a set that parties can
Add elements to the set without revealing the elements or knowing the elements presently in the set.
Proving to other parties that a certain element exists in the set in a way that doesn't reveal at which point it was added.
For example, I want Alice, Bob, Cindy and David to start with an empty set. I want Alice to be able to add 5 to the set without revealing 5, Bob to be able to add 6 to the set without knowing 5 is there or revealing 6, and then Cindy, who cooperates with Alice and has access to any necessary information from Alice, to prove to David that 5 is in the set without revealing whether it was Alice or Bob who put it there.
Is there any protocol that would enable such scenario?