Can set theory be inverted? Has anybody investigated or constructed a set theory, call it $T$, which develops in a manner that is the reverse of how the majority of the standard set theories develop? Instead of starting with the null set, generating the finite sets and expanding into larger and larger infinite sets, either endlessly or until a universal set is reached, the theory $T$ would proceed in the opposite direction. $T$ starts with a universal set and each set (in the universe) of $T$ generates a collection of proper subsets. This descending process either continues endlessly or culminates in finite sets, singletons-and finally a null set. The primitive notion of $T$ might be "$x$ is a proper subset of $y$" instead of "$x$ is an element of $y$" (which is the primitive notion of the standard set theories). Could there already exist such a theory as $T$, that might be of any interest?
 A: One possible way to build sets iteratively from "the universe" instead of the empty set (and the process if fairly symmetrical to the usual one) is to replace axioms with duals. Instead of the axiom of extensionality
$$X = Y \equiv (t \in X \equiv t \in Y)$$ 
use the axiom of "intensionality", according to which, sets belonging to the same sets are equal: 
$$X \sim Y \equiv (X \in t \equiv Y\in t),$$
instead of comprehension-based set builder $\{x\ |\ P(x)\}$
$$t \in \{u \mid P(u)\} \equiv P(t)$$
use, hmm.. "incomprehension-based" set builder $[x\ |\ P(x)]$
$$[u \mid P(u)]\in t \equiv P(t),$$
etc. Shameless plug.
A: Mereology could be described as set theory with the subset relation as the primitive notion. Check out http://plato.stanford.edu/entries/mereology/. The works by Stanisław Leśniewski in the references might contain what you are looking for.
A: This paper - http://www.pgrim.org/philosophersannual/pa28articles/forsterset.pdf - by Thomas Forster describes iterative concepts of sethood which are broader than the classical hierarchy, and spends most of its time discussing an approach which builds both upwards and downwards simultaneously.
