Infinite and Apeiron

George Cantor is credited with the conceptual construction called transfinite numbers, an endomorphization of the concept of infinite within the system of the real numbers (ℝ). The tool for the construction was the creation of a synthetic representation of the infinite as a whole, familiar to the Platonist Weltanschauung, whose condition of possibility was the natural number, i.e., individuation: infinite is then the result of an endless aggregation. Of course such a representation is not the actual representation of an intuition but the representation of a continuing iterative process, which by its closed condition (the finite character of the algorithm of counting), seems to have a meaning. Once the infinite is an endomorphic concept of the system, any composition with it will be endomorphic, including the notion of transfinite number.

Infinite aggregates can be constructed either by endless iterative processes or by the postulation of properties to which infinite extensions must follow. But the aggregation of units is a synthesis of a different sort than the one obtained by the postulation of a property which has some supposed extension of individuals. In the first case, we proceed inductively, and the whole obtains its meaning from the finitude of the algorithm, even if it cannot be intuited in its infinite performance. In the second, we proceed holistically, and the parts obtain their meaning in relation to the a priori meaning of the whole, i.e., the property. In the first case, we reificate the whole, in the second, the units or parts.

When the parts are themselves processes of aggregation, the reification gets of second order and all sorts of paradoxes grow, for now we are violating both the intuition of iteration (iterating infinities), and the intuition of property (the finite representation of a pairing, for we apply the possibility of the construction of the concept to infinite extensions). In this framework, we meet all sort of nonsensical postulates that mathematics forces us to take dogmatically (for intuitively does not follow) under the disguise of theorems that follow from ontoteological axioms (axiom of substitution, choice, infinitude, union), such as: an infinite set has the same cardinality as one of its subsets, a postulate that can only have meaning under the reification of the extension of the set, metaphysically, or the reification of the iterative process in a non-intuitive scale, metaphysically also, beyond all life experience.

From a mythopoetical point of view, the concept of infinity is a representation of the concept of apeiron, whose form is conditioned by the ontoepistemology of the myths assigned to the space-time intuitions. When space-time is curved, infinity is bounded, i.e, finitized under a property. When space-time is linear, infinity is unbounded, finitized under a process of iteration. When space-time is liminal, infinity is just apeiron.