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

Abstract

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 $\mathrm{ZFA}$, 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 $⟨A,\preccurlyeq ⟩$. Next the pre-ordering $\preccurlyeq$ is shifted (by a kind of simulation) to a pre-ordering ${\preccurlyeq }^{+}$ on $\mathcal{P}(A)$, 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 $M_{\alpha }(A)$, along all ordinals $\alpha \ge 1$, the``magmatic 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 $\mathrm{ZFA}$. The class $M(A)={\bigcup }_{\alpha \ge 1}{M}_{\alpha }(A)$ is the ``magmatic universe above $A$.''

24 pages