TITLE: Set theory based on inclusion
QUESTION [8 upvotes]: 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.

REPLY [5 votes]: 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 $\cup x \leq y$.