On 3/14/2013 4:42 PM, WM wrote: > On 14 Mrz., 22:28, fom <fomJ...@nyms.net> wrote: >> 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. > > Of course it is not. He defined, the elements of the set T are > disjoint sets. >> >> Surprisingly, the original 1908 paper introduced >> a union that simply takes a union across the >> elements of the set. > > You do not understand. The elements that Zemelo speaks of are sets. I > give you an example: For instance all subsets of |N are elements of P(| > N).
Obviously, van Heijenoort has deceived all of the English speaking people who depend upon his translation:
"Set theory is concerned with a domain B of individuals, which we shall call simply objects and among which are the sets."
"An object may be called a set if and -- with a single exception (Axiom 2) -- only if it contains another object as an element."
Hmm... Now those *might* be attributable to a poor translation or a misunderstanding on my part. However,
"9. Likewise, for several sets M, N, R, ... there always exists an intersection D=[M, N, R, ...]. For if T is any set whose elements are themselves sets, then according to axiom III there corresponds to every object x a certain subset T_x of T that contains all those elements of T that contain x as element. Thus it is definite for every x whether T_x=T, that is, whether x is a common element of all elements of T; if X is an arbitrary element of T, all elements x of X for which T_x=T are the elements of a subset D of X that contains all these common elements. This set D is called the intersection associated with T and is denoted D(T). If the elements of T do not possess a common element, D(T)=null, and this is always the case if, for example, an element of T is not a set or if it is the null set."
So, what is the important phrase?
"... an element of T is not a set..."
As I said before, one becomes so accustomed to thinking in terms of "pure sets" because that is the modern practice (V=WF, I think).
It is because I pay attention to anything where denotation is an issue that I am so aware of Zermelo's domain description.
The 1908 paper reflects Frege more than Russell. The formal argument for ignoring denotations is probably in Quine's "Word and Object". But, as that is "not mathematics" most are not aware of how it may be significant to the model theory of set theory.