Cardinal Hierarchy

Everybody heard about the famous Kantor hierarchy of infinite cardinal numbers: the next level comes up in considering the set of all the subsets of an infinite set. One could enumerate these infinities with integers (finite ordinals): level-0 infinity corresponds to discrete sets, level 1 is reserved for continuum, all functions over continuum form level 2, etc. Of course one is free to discuss entities beyond this simple layered structure, up to considering uncountable infinite ordinals.

In these terms, the notorious continuum hypothesis states that there is no intermediate cardinality between levels 0 and 1. The statement is very strong, and it cannot be either proved or refuted within the traditional axiomatic set theory. One could observe that this essential discreteness is mostly due to the binary character of the common mathematical rationality, which determines the way we construct power sets. Still, nothing prevents us from extending the idea of a set just a little, retaining the usual notions as natural limit cases. There is a generalization that allows to constructively demonstrate the inappropriateness of the continuum hypothesis in the generalized theory.

Indeed, let us note that the elements of a set in the traditional set theory are joined in the set in an outer manner, as externally opposed to each other. Each element corresponds to the same counting unit, which is infinitesimal for infinite sets, though without losing its qualitative definiteness common for all elements; and this is why we can righteously compare then to each other and count them. Kantor hierarchy therefore provides a series of outer infinities.

Now, assume that the elements of a set are no longer simple counting units, and each element is internally structured. For instance, a single outer space point (an element) could incorporate some inner space characterized by an appropriate Kantor power. In general, the organization of the inner space may vary from one element to another. However, if we are to study some objective integrity, there are good reasons to believe that the mode of unfolding is the same for all elements, so that their inner spaces should be at least of the same cardinality; in many practically important cases (like mechanical motion) one could confidently impose the demand of the same topology.

In the simplest case, the inner space is discrete (similar to the usual spinor components); this is a level-0 subspace. However, each point may also be innerly represented with a continuous area, a kind of zone, which makes it a set of level 1. Such situations are quite common in real life. For example, the perception of a pure tone of some musical height (a logarithm of sound frequency) subjectively pictures it as a distribution of heights in the vicinity of a well-pronounced maximum. From this observation, one can derive that the possibly sets of discernible musical tones form zone structures (musical scales), where each element is far from being a single point, but rather a continuum of the admissible deviations from the center of the zone. What is to be taken for the cardinality of such a hierarchical set? It is certainly discrete, while, on the other hand, it is a union of continuous intervals.

Let us define the cardinality (power) of a two-level set (with a uniform inner space structure) as a pair (K₁, K₂), where K₁ and K₂ indicate the cardinalities of the higher (outer) and lower (inner) levels respectively. In particular, inner space may be absent, which means that it effectively consists of a single element, and its level of cardinality is zero. With all that, a purely discrete set is characterized with the cardinality of (0, 0); a usual continuum has the hierarchical power of (1, 0); a discrete structure with an inner continuum should then be assigned the cardinality of (0, 1). These cardinalities can be naturally ordered in the lexicographic manner, first comparing the upper levels, then the lower levels (if needed). Obviously, (0, 0) < (1, 0). However, as naturally, (0, 0) < (0, 1) < (1, 0). That is, there is a hierarchical set with the generalized cardinality between discreteness and continuum.

No doubt, the process can be continued on and on, as the lower-level elements are considered as complex, in their turn. Sticking to the first two Kantor numbers, the cardinal number of an arbitrary hierarchical set can be represented with a sequence (b₁, b₂, b₃,...), where b_k are either 0 or 1. One can readily observe that this is equivalent to a binary notation for some real number in the interval (0, 1); consequently, there is an infinity of cardinalities intermediate between 0 and 1.

Of course we are free to consider much more complex inner hierarchies. For instance, the hierarchy of infinite ordinals can be reproduced in full. It is especially intriguing to consider all kinds of isomorphism between the spaces of different levels, which will bring us far beyond the trivial tree-like structures, with a number of circularities and loops. Calculating the cardinalities for such sets is a yet another interesting problem.

July 1994