Can the phrase "Something exists" be formalized?

If you try, you would write something like:

∃x∈E

where E is the set of all possible things, defined most broadly.

But if you are trying to use this as an axiomatic truth, a statement that cannot be refuted, then you have to allow the statement itself to be in the set E because otherwise you could deny the truth of the statement. Does that create a self-referential set problem akin to Russell's paradox or is that a legitimate use of self-reference that does not lead to inconsistent results?