Sets with Dependent Elements: A Formalization of Castoriadis’ Notion of Magma

Studia Logica:1-26 (forthcoming)
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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 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 on a set A of atoms (or urelements) can be naturally represented by a pre-order relation $$\preccurlyeq $$ of A with the extra condition that it contains no minimal elements. Then, working in a mild strengthening of the theory $$\textrm{ZFA}$$, 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 $$\langle A,\preccurlyeq \rangle $$. The non-minimality condition for $$\preccurlyeq $$ implies that all sets of $$LO(A,\preccurlyeq )$$ are infinite and none of them is $$\subseteq $$ -minimal. 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 $$\textrm{ZFA}$$. The class $$M(A)=\bigcup _{\alpha \ge 1}M_\alpha (A)$$ is the “magmatic universe above A.” The axioms of Powerset and Union (the latter in a restricted version) turn out to be true in $$\langle M(A),\in \rangle $$. Besides it is shown that three of the five principles about magmas that Castoriadis proposed in his writings (mildly modified and adapted to the needs of formalization), $$M2^*$$, $$M3^*$$ and $$M5^*$$, are true of M(A). A selection of excerpts from these writings, in which the concept of magma was first introduced and elaborated, is presented in the Introduction.

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Athanassios Tzouvaras
Aristotle University of Thessaloniki (PhD)

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