Set theory based on inclusion There are several axiomatizations of set theory based on inclusion rather than membership. I found only two papers, but they are both in German, and I could not read them even using a disctionary. Can anybody refer to any articles in English on this?
The papers in German, which I have, are:
H. Wegel. Axiomatische Mengenlehre ohne Elemente von Mengen. Math.Annalen, Bd. 131 S.435-462 (1956)
A. Schoenflies. Zur Axiomatik der Mengenlehre. Math. Annalen 83, 173-200.
Is there anybody aware of any translation in English of these articles?
What I am mostly interested in, is whether or not in a set theory based on inclusion is possible to express the notion of ordered pair and power set.
 A: See the monograph Algebraic Set Theory by Joyal and Moerdijk for details. Instead of basing a theory of sets on a membership relation, it is possible to take the signature to consist of a unary "singleton" function $s$ and a partial order $\leq$. The intuitive meaning is that $s$ takes a set $x$ to the singleton $\{x\}$ and $\leq$ is intuitively the subset inclusion relation. Then define $\in$ by $x \in y$ if and only if $s(x) \leq y$. It is not difficult to rephrase the ZFC axioms in terms of the primitives $s$ and $\leq$, and in particular one can code up ordered pairs in standard ways. Power sets $p(y)$ can also be defined, and we have $x \leq p(y)$ iff $\cup x \leq y$. 
A: I don't know of any translations, but you could try rolling your own with Google translate. Springer won't give me online access to more than the first two pages of the papers, so I can't try it out on them just now, but here is a slightly tidied up extract from its translation of the Zentralblatt review (by Fraenkel) of the Schoenflies paper:
... As undefined basic concepts and relationships occur: quantity, equivalence, subset, complement of a subset; the union of two sets is introduced by definition, on this basis, whereas that of an infinite number of [sets], as well as the formation of product and power set is not [touched upon]. ...
(Words in square brackets above are my suggested corrections to the Google translation. My German is subbasic, so caveat emptor.)
The Zentralblatt review of the Wegel paper (by Mendelson) is in English, so I leave you to read it for yourself. It is not very encouraging.
