Sets with dependent elements: A formalization of Castoriadis' notion of magma

We present a formalization of collections that Cornelius Castoriadis calls magmas'', especially the property which mainly characterizes them and distinguishes them from the usual cantorian sets. It is the property of their elements to {\em depend} on other elements, either in a one-way or a two-way manner, so that one cannot occur in a collection without the occurrence of those dependent on it. Such a dependence relation can be represented by a pre-order relation $\preccurlyeq$ Then, working in a mild strengthening of the theory ${\rm ZFA}$, where $A$ is an infinite set of atoms equipped with a primitive pre-ordering $\preccurlyeq$, the class of magmas over $A$ is represented by the class $LO(A,\preccurlyeq)$ of nonempty open subsets of $A$ with respect to the lower topology of $\langle A,\preccurlyeq\rangle$. Next the pre-ordering $\preccurlyeq$ is shifted (by a kind of simulation) to a pre-ordering $\preccurlyeq^+$ on ${\cal P}(A)$, which turns out to satisfy the same non-minimality condition as well, and which, happily, when restricted to $LO(A,\preccurlyeq)$ coincides with $\subseteq$. This allows us to define a hierarchy $M_α(A)$, along all ordinals $α\geq 1$, themagmatic hierarchy’‘, such that $M_1(A)=LO(A,\preccurlyeq )$, $M_{\alpha +1}(A)=LO(M_{\alpha }(A),\subseteq )$, and $M_{\alpha }(A)={\bigcup }_{\beta <\alpha }{M}_{\beta }(A)$, for a limit ordinal $\alpha$. For every $\alpha \ge 1$, $M_{\alpha }(A)\subseteq V_{\alpha }(A)$, where $V_{\alpha }(A)$ are the levels of the universe $V(A)$ of $\mathrm{Z}\mathrm{F}\mathrm{A}$. The class $M(A)={\bigcup }_{\alpha \ge 1}{M}_{\alpha }(A)$ is the ``magmatic universe above $A$.’’