Elementary Sets

Most of get acquainted with elements and sets yet in the elementary school, and some may have touched the topic even earlier. As a bit of science still happens to come handy from time to time, the thought of a set seems to be a natural background for further research, before any formalization at all and as its indispensable condition. Something belongs to somebody; some may possess just nothing. This is what we see anywhere around every single day. Well, lets us call a piece of somebody's belongings an element, while the owner of the thing will take the name of a set. It may come up that the presumed owner, in its turn, belongs to somebody else. Attempting to sort that out, we would keep all those who are born to possess within one class, putting those who are made to belong into another class. That is, in addition to sets, there also exist classes, which resemble sets in many respects, but not entirely.

The way we work with sets and their elements is in no way spun out of thin air; it simply mimics the social order dragged from one millennium to another. In sets theory, mathematics is a mere reformulation of (bourgeois) sociology with its principal question: equal or stranger? The attitudes depend on class (estate, race, ethnic, civil, corporate, clan, group) membership. That is why it is so important to decide first on whether the element belongs to a set; the details of the inner structure of the set are in the second line. Obviously, life embraces many things that do not fit in the set-theoretic approach; still, as soon as we have postulated the primacy of the black-and-white logic, the inconvenient questions may well be swept off to a dusty yard side, so that some applied science could take them out for a momentary need, dress up in mathematical gowns and exclaim: how pretty! though the formalities get soon abandoned for something less respectable.

From the set's viewpoint, all its elements are all the same. Of course, they are well distinguishable (otherwise, how could we speak of a set?); still, for some higher judgment, their distinctions just do not matter. The elements, however, may fail to accept such a uniformity: they have something to divide, and some are eager to get more than the others. That is, with all the qualitative homogeneity, there are quantitative differences. The law is the same for all; however, those very big may bluntly scorn it while dealing with the small fry (still demonstrating a pious respect in the relations with those as massive). One way or another, the problem of the difference of equals keeps on, and every generation has to treat it their own manner.

A mathematician goes straight on: each element can be loaded with a number representing the "social weight" of the element, its importance for the whole. If, for example, having a set of two elements {elephant, banana}, we are curious of how many elephants and bananas we really have, and if we find that our bananas much overweigh our elephants, we refer to that set as a set of bananas with an admixture of elephants; conversely, a set with a prevalence of elephants will be called a set of elephants with an admixture of bananas. This exactly corresponds to how the evaluation of people by the size of their capital brings us to the conclusion that this world is made for the rich, while the poor are nothing but a regrettable lapse of nature, a source of petty nuisance. Ideally, one should draw all the poor into some reservation, thus putting them in an entirely different set. Such an exploit, however often undertaken by the political elite, has not yet once succeeded; this circumstance is specifically reflected in mathematics as well.

Why not? Nobody can prevent us from formally representing the community of elephants with banana (so to say, "a mixed state") as a union of two "pure" (one-element) sets:

{elephant, banana} = {elephant} ∪ {banana}

Does that change anything? Nothing, for a superficial eye. However, recalling that an element's belonging to a set is not a heavenly revelation, but rather a statement allowing a practical justification, we observe a very important discrepancy between the two parts of this (meta)equation: in the left-hand side, it is assumed that there is a uniform procedure for establishing the fact of belonging regardless of the quality of each element; on the contrary, in the union representation, two different procedures are assumed, and being an elephant manifests itself in a way other than being a banana; finally, identifying the left-hand and right-hand sides of the equation we make a very strong assumption that one activity will always result in the same outcome that a quite different activity which has little in common with the first. An ocean of question: what does it mean, "a result", "the same", "always"? Different answers, different theories.

Thus, it may happen that the difference of elephants and bananas only exits in the context of the set {elephant, banana}, while an elephant on itself and a banana on itself are undefinable. In a class society, situations like that are quite common: can there be a slave-owner without slaves? a capitalist without wage labor? Well, mathematicians have invented a standard work-around: instead of a set like {elephant, banana}, just take a cortege ⟨elephant, banana⟩, which then can be skillfully manipulated into an "ordered set"; as an immediate consequence, we find that the idea of "order" is to complement the idea of "set", which opens a vast area for proliferating entities up to complete chaos, thus extinguishing any glimpse of reason.

A professional would indignantly repudiate such silly suspicions. Isn't it obvious that a set of two elements can be easily ordered in no time? it is enough to indicate, which element is to be treated as the first. With this consideration, the cortege ⟨elephant, banana⟩ is just a conventional contraction for the set-language record {elephant, {elephant, banana}}.

No, this is not as obvious. What do you mean by "indicating"? There are millions of variants. For instance, we could as well put forth the element with the maximum weight, and order the set in direction from the most "massive" elements to negligible contributions. The cheat trick is in the silent omission of the fact that the set {elephant, banana} is indeed a set, and, as such, it cannot directly become any other set's element. To be honest, we should speak about the set {elephant, X}, where X is, in some sense, equal to the set {elephant, banana}. At this point, any dummy will see that there are so many senses, and the "set-theoretic" definition of a cortege is entirely dependent on how we choose to reduce the complex entities (sets) to simpler entities (elements).

Even for one-element sets, the element elephant is quite different from the set {elephant}. What can be said about sets does not pertain to elements, and the other way round. For instance, we define the cardinality of a set; in classical theory, the cardinality of an element is a nonsense, while the cardinality of a one-element set is unity (by definition). Elements can belong to sets; the maximum what a set can afford is to be a subset (with many pitfalls, here too).

In our common life, we can easily consider the same thing in different aspects, depending on what we are going to do with it. A book can be for reading; but it can also serve as a prop for something, or become a kindling stuff; a book can incite hostility, or, conversely, be a symbol of unity... In the same manner, anything can be treated sometimes as a set, and sometimes as an element; but these still remain two different treatments! An arbitrary mix of the both in the same discourse would be a logical fallacy. Over and over, we meet the same problem: what is "the same"? No formal theory can answer; this is an entirely practical decision. One could easily observe that, with any idea of the unity, the elementary and set qualities will be its different aspects, the distinct modes of usage. Such partial manifestations may be almost independent of each other (thus, one could be a good dancer, but also a poor husband); in reality, we much more often encounter the opposites that are essentially interconnected, up to being utterly incompatible, so that defining one of them, we also define its counterpart. What is taken for an element is not a set; what is a set cannot be an element.

Can we fancy something that would be both an element and a set? Yes, we can. But that would lead us beyond the limits of the classical set theory to consider all kinds of the superposition of "pure" sets, or some analogs of the quantum-mechanical density matrix. That may open a most promising direction of research, a different science.

Meanwhile, let's turn once again to the sets whose elements, beside the qualitative definiteness, may also have a quantitative load, a weight. It might look like a trivial generalization of the classical set, where the possible values of the weights are restricted to zero and unity. Indeed, let us admit that some elements exist as several instances... To exhaust a set, we just blind-hand rummage in a set and pick some of its elements; let it be WR sampling well known in mathematical statistics, or repetitive enumeration like in computer programming (with its bag objects and various iterators). Nothing new.

But hold! Sets with repetitive elements essentially differ from "weighted" sets: in the former case, we deal with the classes of equivalence; in the latter, with the intensity of the presence of each element that cannot be divided into separate instances. An exhaustive enumeration of the first kind will give the complete number of all varieties (with an exact quantity for each variety); the enumeration of a "weighted" set will give each element just once, but with a certain weight (in one of the possible modifications). Classical and quantum statistics differ in a similar manner.

Well, this too does not seem to be much of a news. We have long since invented the theory of fuzzy sets (though, in fact, it still remains a good intention rather than true theory; a template for special implementations depending on the techniques of combining the membership functions). The idea is really great: an element does not entirely belong to a set, but rather tends to belong; in particular, one comes to distinguishing the volume of a set (the number of elements) and its mass (the sum of the weights). In principle, the weights can evaluate to any real numbers; however, we are free to normalize them to the total mass to bring the weights to the interval (0, 1), or, alternatively, divide the weights by the number of elements, which means a transition form extensive quantities to densities (so handy for infinite sets). The wider conceptual choice, the better for applications.

Now, what is missing in the soup? Just one thing, science proper. Those who have learned to press the computer keys or tap the screen cannot yet be considered as IT gurus, or at least power users. The monkey mode of working with sets does not make the matter a trifle clearer: we still do not know why we should act that very way, and where we'd better seek for a different toolkit. To spread like must over a volcano is far from specifically human ways. Reason is to make the knowledge of a science's limitations grow along with that very science.

Mathematicians are apt to believe that formal constructions exist as they are, given to the humanity from a superior realm, so that the earthly things can only implement the universal ideas in a random and incomplete manner. Hence the usual terminological confusion: in mathematics, the term "model" is used to denote special implementations of the abstract schemes, while reality is exactly the opposite: mathematical theories are nothing but vague, one-sided, approximate reflection of certain aspects of human activity; one need to add much and much to mathematics, to make a formal carcass of a real thing just a little bit meaningful.

Most often, formal knowledge is just a kind of covertly acknowledging one's ignorance: we know as much as nothing, but we can well make up for it blowing the cheeks and pretending to be wise. Let a stranger ask: why at all? and there is a prompt answer: this is an old and respectable tradition! We are used to do that, and those who dislike it may proceed their own way, we don't care. Well, the mere fact of a conscious choice of the mode of action is already a good sign. However, there is nothing bright in stretching (or cutting) the world to habitual standards. As the world would actively object to such a treatment, the choice turns into self-isolation, rejection of anything that is not our way: true science is just what we do, and those who think differently cannot enter the academic circles (with the appropriate organizational and financial consequences).

Science begins with trying to name everything, to move from working with things to playing with their abstract representatives, terms and formulas. It's quite normal; however, science should not stop at that. For example, in medicine, many diseases are referred to by the name of the organ that does not properly work in some respect: gastritis, bronchitis, periodontitis, sinusitis etc.; the next step is to find out what exactly goes wrong, to be able to further ask why. Mathematics is primarily a universal technology of making names. What stands behind the names is to be clarified at the next stage, with more appropriate methods.

Inventing a name is in no way going to augment knowledge. Naming is an important preliminary phase of cognition, a kind of a declaration of intent: we have noticed some peculiarity in the world, and we are to look closer and dig deeper. All right, we can fiddle about with sets; now, let us pay attention to what we really do.

In the general philosophical plan, each conscious activity has its object and product, and the subject of activity is to transform the former to the latter. In any particular activity, nature is first perceived as a chaos of things and events, the possible objects. We need to select from this mishmash something that could help us in pursuing our goals. Each dish requires certain ingredients; we seek for them in our environment, ticking off the needs that are already satisfied. When all the boxes are checked, we can start working, as we have at last the specific object of that very activity, the raw material for the intended product. Such objects are represented in mathematics by sets.

This already implies much. To become an element of a set, a thing must feature quite definite (in the current context) properties, that is, match the industrial standards. When a taskmaster makes a call to the construction site asking whether they have nails, he does not needs a formal answer (yes, I still have a couple in my pocket) but rather an estimate of the sufficiency of the available stock for a particular task. In other words, the membership of an element in a set means the completion of one operation (acquisition) and switching to another (production proper).

Further, since the list of ingredients is determined by the end product rather than the conditions and circumstances of activity, sets can be constructed in many different ways. In the simplest case, we have something like a number of measures that are to be filled with the appropriate matters (one egg, 200g of flour, 100g of sugar, a pinch of salt, baking powder on a knife's tip, 1/4 glass of cream). When we fail to meet a particular request, we can use something of the kind, respectively changing the proportions. That is, each product determines a number of sets rather than a single set; this is a set-theoretic universe with a structure adapted to (and formally studied within) a specific problem. Instead of a single set theory fashioned once and forever, there is a bundle of set theories induced by special formal models (mathematical products). The degree of resemblance of one such theory to another is in no way related to their inner virtues; it comes from the similarity of the parent activities.

Philosophy indicates that an activity (as a manifestation of reason) cannot be a mere incident, a momentary and unique act; any activity is a cultural phenomenon implying regular reproduction of some product and the social conditions required. As a consequence, the phases of production preparation and production as such do not need to follow each other in a strict order: in the cyclic reproduction they become relatively independent, so that we can just accumulate everything needed for production and spend a part of the stock when all the prerequisites are in place; the rest is to be passed to the next production cycle. This exactly what the mathematical abstraction of a "weighted" set means: a "store", a number of cells for raw materials, plus the level of occupation for each cell. Quite naturally, a lot of possible implementations. Thus, when the containers are vast enough and the ingredients do not significantly interact with each other, one cell can contain as many items as we like; in the opposite case, one cell can contain just a single item (or be empty). This is the way the "boson" sets differ from "fermion" sets (the latter constitute the realm of the classical set theory). And, of course, any conceivable combinations.

An important margin note: the quantitative differences of the elements are related to reproduction cycles; in a static theory, they represent time. Mathematics will always incorporate these two aspects: qualitative definiteness (spatiality) and enumeration (sequence, ordering, as an abstraction of time). One can be expressed through another as largely as we like; this won't remove the very opposition, rather shifting it elsewhere, to some hidden suppositions. The unity of space and time is only possible in a special activity that, for our purpose, could be called measurement. All of a sudden, our modest industrial warehouse would grow into a full-fledged space: a spatial dimension corresponds to each element, while the weights of the elements turn into spatial extent along the respective axis. In this picture, the traditional, classical sets are represented by various hypercubes; less trivial constructions may refer to spatial areas of an arbitrary form. For an alternative, one could prefer labeling classical sets by the points of such configuration space; this gives room for other generalizations: for instance, we could be interested in the dynamics of transforming one set into another, of course, within the same matrix activity, in respect to its product.

That is not yet the whole story. Classical generalizations of traditional sets assume a quite definite mode of establishing an element's membership in a set: if we put something in, it is expected to stay there for all times (or, at least, until the next scheduled inventory). In other words, the elements of a set are believed to be well isolated from each other, they never react. Which is certainly not the common case. Real things go off, produce other things, change location, merge and split. Put dough in a stove, and take bread out; plant a seed, and get a prolific tree. Additionally, the very way of extracting a thing from the stock may involve various transformations: thus, having bank accounts in US dollars, euros and Russian rubles, we may wish to get some Suisse francs in cash. This is how quantum sets are born.

Specifically, a single numeric estimate of an element's membership in a set gets split into complimentary parts, just like quantum mechanics begins with replacing real probabilities with complex "amplitudes". In Dirac notation, the possible incoming elements are represented by the vectors of some configuration space |α⟩, while the available outcome of any sampling is described by the functionals ⟨μ|, so that obtaining the "membership function" μ for a set in the state α is related to the transition amplitude ⟨μ|α⟩. The inherent activity-framed character of a set is vividly stressed in the so called second quantization formalism, where adding an item a means application of a creation operator a⁺, while removal of an item form the stock is related to an annihilation operator a^–. Just like in quantum mechanics, the operators may be noncommutative, so that the structure of the set would depend on the process of its construction. When mathematicians boldly identify the objects obtained in different ways (2 = 1+1 = 3–1 = 12/6), they implicitly presume the existence of some activity involving all such products on the same footing; there is no random choice or arbitrariness, this is an entirely practical issue. Without such a cultural background (albeit in the form of children's play or abstract curiosity), there is no science at all, but rather (in the most innocent case) an art of (symbol or public opinion) manipulation.

Now, from the viewpoint of the matrix activity, a set is just a number of tags, or labels, that we stick onto anything at all that might have some (however distant) relation to the job. In psychology, such a categorization, grouping the observed phenomena according to pre-determined criteria, is known as perception, which differs from mere sensation by that very universal activity of triage. Sensation would produce an image of a thing as it is, and we (as any other living creature) are made so that such an adequate reflection were possible, of course, within certain limits. Sensory pictures may be insufficient, but they never fool us. On the contrary, perception gives the image of a thing as we picture it to ourselves; here is a source of illusions and mistakes. Historical development, in this respect, is to compile all kinds of conceptions suited each for its specific task. Ideally, in any activity, we should employ the gauge that would reproduce the organization of the activity as neatly as possible. In real life, any correspondence is but approximate; still, some choice of a reference frame is inevitable, since there is no other way to outline the object of activity in the infinitely diverse world. In psychology, the perceptive scales like that are called sets; the same holds in mathematics.

In this context, it is evident that the formal constructs like an "empty set" or the "set of all sets" cannot be proper sets; they belong to other levels of activity and their object are is different. Our perceptions are culturally determined; hence our conceptions are not arbitrary, they always reflect the already available practical options. Conversely, each object area is culturally linked to certain types of activity. With all that, the idea if an empty set may refer to the object area, while the hierarchy of the possible scales provides a common universe embracing sets as its "members"; still, a set as an "element" of a universe is not the same as that very set as a collection od elements, and it would be logically incorrect, to mix the two connotations in the same formal context.

The notion of set may seem to be more general than the notion of a "weighted" set of any kind: in formal consideration, we apparently "superimpose" additional characteristics on a carrier set, just attaching the weights (population numbers) to the original elements. However, the same formal approach would say that, conversely, a set is a special case of a "bag", one of its possible projections. The right answer is that the both abstractions reflect certain levels of activity; in every hierarchy the order of levels is relative, it depends on the way of unfolding. No doubt, in some cases, it is quite enough to think of a set as a collection of boxes that can be filled or emptied. Reality is often different, and the structure of the "store" may follow the evolution of the industry, adapting to the current needs. For example, take the different classes of cargo vessels: tankers, bulk carriers, container ship etc.; similarly, the invention of the virtual circulating media will significantly change the currency market. We are free to recourse to any abstractions, provided we do not forget about their limited area of applicability. The interrelations between abstractions never refer to inherent superiority of some of them above the others, but rather to their common origin from something that cannot fit in any abstraction at all.

In the spatial picture, a union of sets corresponds to an increase of dimensionality; the intersection cuts out a common subspace. Still, from the objective viewpoint, in both cases we speak of kindred activities with a common object area. However, such a commonality can be sought for in two opposite directions: either this is an extensive search accumulating everything that has ever been used for something, or one could seek for universality, selecting what might feed many similar products. One always goes with another: many different things must be present to reveal kinship, and there is no distinction but within a certain commonality. Formal science is often seduced by the mirage of abstract universality: just find the most fundamental building blocks of the whole world, and there would be nothing to wish for, and the humanity would cut any worries and rest on the laurels... However life punished people for that lazy complacency, some scientists still believe in the final law of all that happens: well, in the early epochs, we made lots of mistakes, we did not properly understand, but now, with all the fruits of the progress at hand... Eventually, this does not differ from religion, up to the sign flip. Real world is subject to change, and not only due to spontaneous (background) evolution, but mainly because of our conscious interference, in the course of cultivating and intentional reformation. Every now and then, we put ourselves in a different environment carrying the traces of our influence on natural things; this adds yet another level of mediation between the properties of things as they are and their cultural involvement.

New technologies lead to new sets, be it aircraft construction materials, or kitchen stuff, or inhabitable lands. Climatic shifts will modify the meaning of the seasons and their duration. Some of the terrestrial languages are bound to die out, and some new languages are yet to come. Similarly, the units of measurement in physics and engineering get adapted to the new variation range. It is most unlikely that mathematics could always stick to the millennia-old mental framework.

Nevertheless, many cultural phenomena persist for long, compared to the average duration of the human life, or even to the span of a single historical epoch. Within that period of stability, as long as the overall character of activity remains the same, there is no sense in proliferating theories; any observable modernization is a manifestation of a hidden social need and a presage of a revolution. Of course, nobody can just cancel earlier habits; they will coexist with the new trends in several generations.

Universal sets can only exist as a complement of non-universal sets, which, in their turn are not universal only in respect to some earlier established universality. Under certain conditions, one opposite can easily transform into another. This is what we call hierarchical conversion.

There is a popular candidate to the place of a universal form: language. Apparently, as long as we restrict ourselves with the cognitive aspect, language is a true embodiment of the idea of categorization: every distinction has a name, so that the vocabulary becomes a natural universe for everything. Yes, the vocabulary is bound to expand; but these are regular modifications that are quite compatible with the ideal of a super-dictionary containing anything at all, and serving as a natural limit for every linguistic development. The next step is to canonize the alphabet; the makes the limit an absolute common for all minds, and the academic nirvana seems not so far away...

This line of thought piles one illusion upon another. Language cannot evolve in a purely quantitative manner; primarily, its development means a change in its content. Even within mathematics, huge conceptual changes have occurred in just a few centuries; as a result, old texts in the modern interpretation may be far from the author really meant. The same holds for the presentation techniques, the "alphabet": the zoo of mathematical notation does not fit in Unicode, virtually loosing the very sequential appearance and employing many-dimensional diagrams that cannot be reduced to the traditional discourse but in a few very special cases. The words "number", "space", "function", "truth", and of course "set", get re-interpreted by every generation of mathematicians in comparison to the terminological newcomers like "algorithm", "applicator", "topos", or "fractal".

Taken in different proportions, the elements of a set determine different products; all that unfolds on the base of a definite technology, so that the possible component structures would lead to compatible (or comparable) products. To pass to an entirely different class of products, the very object of activity is to be changed; in general, such objects do not depend on each other: to allow for their natural union, we need an activity somehow embracing them all, with all their specificity. That is, some activities may implement an activity of a "higher level" (see above about the relativity of such an ordering). The hierarchy of sets is to formally reproduce the hierarchy of activity. In the simplest case, we get a tree-like structure: food industry is pictured as a number of relatively independent branches like agriculture, dairy industry, fishery, cattle breeding... Om the other hand, any recipe would combine quite different ingredients like flour, eggs, cabbage or meat; with al that, bread is something different from a fish soup or an ice-cream dessert. This, once again, illustrates the flexibility of hierarchies, heir ability to unfold in many hierarchical structures without loosing the qualitative definiteness. Nothing to say about more intricate inter-level relations far from a trivial tree.

With all these considerations, set theory (elementary or not) can be explored in a cultural context permitting most diverse generalizations. This is the only way to make mathematics truly meaningful. Everybody is free to play with forms; moreover, practical applications do not necessarily involve sheer material interests, as we can be inspired by beauty, be systematic, or dream of the future. Any need demands certain instrumentation, and formal instruments in particular. Because of the unity of the human culture (as an expression of the unity of the world), no theory, however weird, can grow from nothing or be utterly useless. Still, an extra bit of reason in our attitude toward our creativity will hardly do any harm.