Objective Set Theory

The traditional set theory says nothing bout the elements of a set. Admittedly, sets can become elements of other sets, without losing any of its set qualifiers while treated as an element. Such an approach suffers from too much generality, and the indeterminacy of the basic notions may lead to all kinds of paradoxes. To overcome this difficulty, mathematicians have already weakened the original universality and come to considering classes as different from sets, and less restrictive. The next logical step would also introduce the qualitative difference of a set and an element. One cannot substitute elements for sets or sets for elements; mixing elements and sets within the same argument (or formula) would be a logical fallacy.

In real life, any science is only applicable within its object area. Its abstract instrumentation will only mimic the real organization of the object. Otherwise, the theory is utterly nonsensical and of no practical use. Applied sciences do not need too much rigor: affordable recipes for everyday tasks are much more valuable. Even considering very different application, with some of them retaining a very high level of abstraction, we do not eliminate the obvious fact: there is something to study.

So, why not explicitly lay some object area in the basis of a theory? For mathematics, the particulars do not matter; it is only important, that there are somehow organized objects, and one cannot arbitrarily introduce imaginary entities. For a set theory, the object area provides some universe U, the ground level of the theory. On the next level, we get sets proper, as collections of objects from the universe U. In contrast to the traditional approach, sets cannot immediately belong to the universe: they are objects of a different kind. A one-element set {x} is not the same as the element x. The relation of containment a ∈ A or non-containment a ∉ A connects two adjacent levels of hierarchy. We can employ the usual notation for relatively small sets, just listing its element if the braces: {a, b, c, ...}. All thus included elements belong to the universe and cannot be sets. With this distinction, there are no problems of set formation by a common property: when the elements possess an objective property φ (compliant with the nature of the object and the logic of the theory), they can be taken together to form a regular set. The notation {x | φ(x)} is acceptable for any objective properties; moreover, each set will determine some objective property as common for all its elements. When elements are clearly separated from sets, reflexive conditions like x ∈ x are all out of reach.

Generally speaking, not all the collections of objects from the universe U will represent admissible sets. Different object areas may impose their specific restrictions. Without such constraints, for each individual object x from the universe U, there is a one-element set {x}. For a finite universe one could speak of the set containing all the objects from the universe; however, in general, such a "universal" set may be unavailable. For example, some objects may be not present in U all the time; also, some objects may have mutually incompatible properties and hence they cannot be contained in the same set. From the very beginning, we assume that every set contains at least one element; the common idea of an empty set ∅ = {} may only appear in an object set theory in a metaphorical sense, as an abbreviation for the phrase "there is no such set that..." We say that, by definition, set is something with elements. When there are no elements, this will refer to an entity of a different kind, to be studied separately.

If one set is equal to another (A = B), they are the same set, just differently labelled, as alternative construction paths may result in the very same collection of objects. The equality of sets is always understood as objective equality, the same property of real things.

One set can be a (proper) subset of another: A ⊂ B. Then we speak of a more specific property, narrowing B to its part A. Note that its is real objects from the universe U that are meant: some of them belong to one set without being contained in the other. Regardless of the number of elements (cardinality) the set B will be wider than the set A, if there are elements of B that do not belong to A, but not the other way round.

On the set level the usual definitions of set union A ∪ B and set intersection A ∩ B will hold, as well as the complement of one set to another B \ A. However, the formal construction does not tell us anything about the existence of the resulting set. Thus, the union of two sets may fail to exist in the presence of certain constraints, when some elements are incompatible. Since there is not empty set, the intersection of sets must contain at least one element; otherwise we honestly admit that the sets do not intersect (are disjoint, or disconnected). Similarly, the complement is only possible for a proper subset: A ⊂ B. In other words, the availability of any set-theoretical operations is determined by the nature of the object; conversely, we can judge on the structure of the universe, on the basis of the available set structures.

The collection of all the implementable objective sets can be treated as a higher-level universe, to construct the "sets" of sets; in respect to the object such constructs are classes rather than sets; that is classes are entities of a different level which they cannot contain objects from the universe and cannot be directly derived from the universe.

While set construction mainly refers to the objective properties, classes are more appropriate to convey the logic of the theory, the way of treatment and interpretation. The different theories of the same can be built using specific principles of class formation.

In the simplest case, when all the combinations of properties are possible, the class level exhibits a remarkable structure. For each one-element set, we can consider the collection of sets intersecting with that core set. In other words, we consider all properties compatible with a given object from the universe as a class, the collection of sets containing this very element. Since every set belongs to the class of any of its elements, the class level is entirely spanned with the above elementary classes. That is, classes can be uniformly mapped into the underlying universe, so that the level of hierarchy seemingly merge. This is known as hierarchical conversion: an element belongs to a set; but that set, in its turn, belongs to the corresponding elementary class. This three-level scheme represents here a formal theory, a mathematical model of an object area.

There is yet another direction of unfolding the hierarchy: any given set can be mad a universe (base) for the sets of the next level. Such sets could be called internal, as compared to classes ("external sets"). Internal sets obviously correspond to the subsets of the base; still the two levels cannot be identified, as elements cannot be directly compared to sets. One can also observe that the collection of internal sets is limited by the available subsets; this may serve as one of the possible definitions of a constraint. The classes of internal sets are constructed in the same manner. With the rich enough original universe, very complex hierarchical structures could be developed.

Now, let us take a couple of universes U₁ and U₂. Each of them will produce its own set-theoretical hierarchy. The components of these hierarchies cannot be immediately correlated, even at a similar level. This would be like arbitrary addition of millimeters with kilograms. The build a unified theory, we need to join the two universes in one, and then construct any higher levels. As usual, such a unification can be achieved in many ways. The two principal paradigms are given by sequential and parallel linkage, corresponding to time and space, the inner and the outer.

The sequential synthesis will produce a universe that is known as a (direct, or Cartesian) product of the original components: U = U₁ × U₂. Each object from such a joint universe will be an ordered pair of original objects form U₁ and U₂: x = ⟨x₁, x₂⟩. This is important, which of the two component universes is taken as first, and which follows. Even in the case when the two universes coincide, they will enter the direct product each in its specific quality, as the first and the second. Any of these positions may imply its own constraints, and there may be oriented constraints limiting the number of the admissible pairs. The Cartesian-squared universe U² is different from mere numerical power; in particular, in each pair ⟨x, x⟩, the object x in the first position is different from the object x in the second: the same object is taken here in its different aspects. For example, it may be taken at different time moments, significantly changing from one to the next. Of course, formally, we might adopt the inverse order as well. However, this will result in a different formulation of the theory. Thus, instead of x₁ "precedes" x₂, we will employ a different terminology, saying that x₂ "follows" x₁. The synthesis of the universes is objective; it implies a practical usability of the joint universe, the real distinguishability of the positions in a sequence.

In this mode of synthesis, sets will contain various ordered pairs, accounting for the imposed constraints. Every element of a set includes the components of all the incident universes. Obviously, the existence of universal sets for both component universes would mean the possibility of treating the sets over the joint universe as subsets of the ordinary Cartesian product of sets. We know, however, that such global sets may do not necessarily exist, while the Cartesian products of the component sets do not always produce the same pair collection, especially in the presence of oriented constraints.

Sequential synthesis corresponds to considering different aspects in the same object, which are relatively independent and can sometimes be studied separately. For example, the temperature and pressure of a gas, the width of a river and its depth, the same society in different epochs. On the contrary, parallel synthesis joins one universe to another in an outer way, as two "parallel" realities (for instance, the disjoint areas of the same space, or the components and phase states of physical mixtures). In this case we speak of a (direct) sum, or a superposition: U = U₁ + U₂. This sum is commutative, as we are interested in the very presence of the object rather than their ordering. In well-developed hierarchies, such superpositions may be formed with some weights on the component-universe elements. Here, the objects from the two components coexist in the same moment, and we can employ all of them. Then every set A of the joint theory will be representable as a direct sum of the component sets separately produced by each universe: A = A₁ + A₂. This differs from the usual set union by that the two components never mix up, independently participating in any set-theoretical constructions. In particular, one cannot speak of the overall number of elements, but rather retain two numbers, one for each component. Certainly, some combinations (direct sums) of the sets over the incident universes U₁ and U₂ taken separately may be absent among the sets of the joint universe; the existence of the result is to be established in every particular case, accounting for the imposed constraints.

In respect to the elements of the sets, sequential synthesis could be considered as inner junction: every element of a set will be split in two components. Parallel synthesis acts in the opposite manner: the sets are taken as a whole and joint together without influencing their elements.

If A = A₁ + A₂ and B = B₁ + B₂, the union of the sets can be defined as component-wise: A ∪ B = A₁ ∪ B₁ + A₂ ∪ B₂. Since there are no empty sets in an objective theory, every set above the joint universe will contain the both components. This is quite like a Cartesian product producing pairs of objects where there are the both elements, so that the positions in a cortege cannot be empty. That is why the product A ∩ B = A₁ ∩ B₁ + A₂ ∩ B₂ exists only when there are common elements in the sets A₁ and B₁, A₂ and B₂, pairwise. Note that neither of the component intersections must necessarily exist in the hierarchies over U₁ and U₂ as separate universes; this does not prevent them from appearing in a composite set.

In general, the universe U can unite many components, with different junction types. In such hierarchies, it is especially important to complement formal constructs with an analysis of the existing constraints. The correctness of judgment, in such a theory, depends both on its inner logic and the nature of the object. Let S be, say, a universe of symptoms, C will denote the individual contraindications, and M will be the universe of therapy schemes. Then the sets over the composite universe (S + С) × M will represent the possible instances of medical treatment. Obviously, only a part of the possible combinations will make sense. For a constraint, one could take the overall dynamics of the disease, which is to be directed to the final recovery, rather than the other way. The specific dynamics of treatment will be representable in this scheme with a sequence of sets, to account for the possible changes in the patient's state.

For the sets over the composite universe, internal sets and classes (external sets) are define the usual way. Here, beside elementary classes, we will also get all kinds of projections, that is, the classes of set coinciding in several components. For instance, consider the parametrical family of sets A = A₁ + *₂, where * stands for an arbitrary set over the universe U₂. The class of all existing over U sums of that form (the projection onto A₁) determines a class over U₂, which could be called adjacent to class A; in the general case, such a class may be absent in the standalone hierarchy over U₂, and its intersection with the native classes of U₂ (if existing) will form the boundary of the family A in U₂. Similarly, the classes of Cartesian projections can be defined as the families of sets with the elements of the type ⟨x₁, *⟩ or ⟨*, x₂⟩. As for elementary classes, the whole hierarchy becomes reflected in itself, since thus obtained structures can be related to some of the already introduced constructs of the theory.

Of course, one is free to consider any other (logical) structures on the set level with the corresponding "classes of equivalence". Still, arbitrary formal constructs won't automatically become components of the theory; they yet need an objective interpretation. The correspondence between the classes and the objects from the universe can serve as a criterion of truth, limiting the applicability of formal operations.

To resume: every objective set theory 1) requires some (not necessarily formally definable) universe; 2) builds a set level over the universe and considers set classes; and 3) identifies the classes with the objects from the universe. The arrangement of the theory will thus reproduce the organization of the object.

Aug 2006