On 3/14/2013 2:58 PM, Virgil wrote: > In article > > If that is what Zermelo said then he was wrong to say it because one > does not ever "union" elements, only sets.
I want to say that is technically incorrect.
Surprisingly, the original 1908 paper introduced a union that simply takes a union across the elements of the set.
That is, a union consists of the elements of the elements of the given set.
I say surprisingly because Zermelo admitted objects different from sets in the description of his domain. That definition of union would mean that two sets could have the same union if they contained the same sets but different urelements.
Of course, the axiom of union is forced to take that form. What I mean is that it may be interpreted as "closure under arbitrary union" where arbitrary applies to whatever exists in some given model.
It is just that one becomes so used to thinking in terms of pure sets.
And, naturally, your mention here applies to casual usage just as we do not write proofs in formal derivations.