Communissima or transcendental collections are not numbered

The ontological problem of the many is understood as an open-ended collection type which is known since antiquity as a transcategorial, communissima or transcendental collection. It is argued that the extension of communissima predicates such as something(x), one(x) or being(x) is a non-numbered collection. Instances and occurrences of one and the same x are distinct somethings: there is no automorphism of the communissima collection over itself showing its equinumerosity. Non-numbered collections are cardinally wild or indefinite, but they are not necessarily epistemically or semantically undetermined. The main significance of un-numbered collections and particulars is their a-numerical identity. Un-numbered collections and particulars are themselves, but they are not numerically identical to anything, including themselves.

Unger’s problem of the many , Putnam’s problem of the number of objects and Sosa’s explosion of reality share the same foundational problem as the open-ended domains of sets posed by Fine and Williamson on irrestricted quantification . The transcendental notions seem to be bad, open-ended, undetermined sortals. Thus, one(x) produces the distinction between the unum numeri, and the unum transcendentalis . Avicenna, Aquinas and Suárez offered distinct lists of transcendentals but did not distinguish between infinite and indeterminate quantities as contemporary logic does. We may now use contemporary tools to capture the communissima intuition and study if the extension of predicates—when interpreted over anything whatsoever—is numbered or not. The core of the problem is abstract but simple enough: does an extremely general collection allow an automorphism showing its equinumerosity with at least one collection (in particular, with itself)?

We shall call any quantity or amount of x unnumbered or “wild” if there is no cardinal number of x . An unnumbered collection does not have 0, 1, n, w, k or any finite or infinite number of elements . Let x be a value in the extension of the predicate something(x), one(x) or being(x).
Let us assume that context does not restrict in any way the range of these predicates and they are genuinely common to everything. The nature of x is irrelevant. As an arbitrary value of a variable, x satisfies the corresponding propositional and categorical logical axioms, possibly including = axioms. Notice that these axioms do not imply the numerical identity of x with itself.
The communissima collection is not numbered as a result of these two principles:
- Instances and occurrences of x are something
- The numerosity of x depends on x and not on its instances or occurrences
Thus, even if the numerosity of x is one (a singular particular), the number of its instances and occurrences in the communissima collection is indeterminate. The collection of all something/one/being does not allow an automorphism because its image is another distinct something/one/being. An instance, copy or repetition of the same x is the same identity, but a distinct occurrence, and hence a distinct something/one/being.

Since the communissima collection is not numbered, something is not numbered at least in some models. Notice that if a particular x is unnumbered, then x is not numerically one, and therefore not numerically identical to anything. In consequence, the a-numerical nature of some realities is a candidate solution schema for the aforementioned metaphysical difficulties.