# 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.

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 $\cup x \leq y$.

• Ah, that's nice. It seemed clear to me that $\leq$ by itself isn't enough, but I was imagining much more complicated schemes for adding the needed functions to make this work.
– user14972
Oct 9, 2013 at 23:36
• This approach is very attractive, and I previously thought that singleton function and inclusion are sufficient to serve as primitives of a set theory. Unfortunately category theory language in which this is presented in the book is difficult for my intuition. I also found a book David Lewis "Parts of Classes" formulae, but I cannot put down any axioms of this theory from this "too invformal book". Is there another source where the axioms of this theory are given? Nov 13, 2013 at 16:46

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.