Crazy Kantor

Since the earliest school experiences, we know that numbers can be either natural or real. All the rest (rational, complex numbers, quaternions, octonions, transfinite numbers, ordinals etc.) can only be called numbers in some less obvious sense, as they basically appear as specific structures upon the true numerical foundation (probably with some recourse to topology). Even real numbers are indeed a source of doubt and confusion: the speculations about the limit transitions in the bundles of rational numbers can hardly convince anybody with a basic mathematical background, as the thought of a final result of an infinite process seems to be an apparent logical fallacy, a kind of wrong qualifier. As it is well known, real math came to support the everyday activities (like land measuring, comparing volumes and masses). The relation between real and natural numbers is therefore quite similar to that of the measurable and the measure, an existing thing and its perception. And these opposites are certainly distinct. We are well aware of the fact that our notions of the world are necessarily limited, restricted to its minuscule portion that needs a practical effort here and now. There are hidden motives, and there are explicit goals. As well as an ocean of whatever is beyond the scope of a particular activity. Hence the idea of discreteness and continuity, which are irreducible to each other just because they refer to different object areas that cannot be compared in a straightforward manner.

On the contrary, the academic mathematics accepts from the very beginning that there are no constructs differing from each other to any significant extent, so that their interrelations are purely formal, and one thing can always be reduced to (or deduced from) another. The unmeasurable richness of nature is thus squeezed into something flat and barren. The mathematical picture of the world is extremely rough, approximate, incomplete. Such immodest abstractions still can be useful sometimes. However, the generality of reduction and its imperative ubiquity are sheer illusion. Logical faults and the vagueness of the fundamental principles are the price to pay for unwarranted ambitions, blind enthusiasm and pretentious exaggerations. As the other side of the same, too much scruples about the form and the standards; in the civilized society, the ostentatious courtesy and dress code serve as a common disguise for mutual hatred and instinctive vulgarity.

As soon as we drive out the distinctions between integers and real numbers, there is the question of the very their existence: what are they? do they refer to assumingly different constructs, or are they merely two representations of the same? The education standard insists on the traditional answer: there is absolutely no way to map natural numbers to reals, and the other way round. This is what the great Kantor has demonstrated with all the possible rigor. Which discovery has brought him in an asylum for the rest of his life. To be sure, the equality of infinities is to be commonly understood as isomorphism, the revertible mapping of one onto another preserving the essential properties of the both.

This latter formulation is already a prompt target for skepticism. In Kantor's epoch, even the greatest mathematicians would not unanimously vote for the actual existence of any infinities. What is a mapping of one infinity onto another is no more obvious. Nothing to say about the criteria of importance of any properties. To overcome the lack of harmony, mathematical philosophers have invented a peculiar mode of speech allowing to discuss the incomprehensible in an entirely formal tone, regardless of the sensibility of such a science. This obligatory platform proudly bears the name of axiomatic approach, and every person of any age and race must conform to the standard on penalty of permanent excommunication from mathematics, and science in general. Still, sweeping the litter under the carpet (or the avoidance of the thought of a white monkey) can never withdraw any real problems. And logical problems in particular. Those that make the allegedly rigorous mathematics utterly non-convincing.

Let us recollect the textbook-haunting diagonal scheme of that Kantor's proof. First of all, real numbers (conventionally restricted to the segment [0, 1]) are to be identified with the sequences of binary digits, 0 and 1. We are told that every sequence of that kind represents a real number in the binary positional notation, and that every real number is therefore representable in this form. Well let us skip the questions about the admissibility of this two-way logic and leave aside the very nontrivial nature of the positional notation (or a "fundamental sequence", as Kantor put it), as well as the issue of the identification of a thing with its name. Just note that the very possibility of matching one infinite sequence against another is inherent in the very mode of problem formulation. Now, assume that all the real numbers can be enumerated, that is, put in correspondence to the elements of the naturally ordered naturals. Once again: as such an enumeration is infinite, there is an issue of the feasibility of establishing the correspondence between a binary sequence and its number; so far, we voluntarily suppress the discussion. It is also assumed that, for the sequence labeled with any natural number n, it is possible to determine the value of the n-th digit (which will admittedly be either 0 or 1). One can easily revert this value, taking 0 instead of 1, and 1 instead of 0. That is, there is an exact correspondence between the number n and a binary digit, which gives a binary sequence that cannot coincide with any of the earlier enumerated sequences, since (by construction) it differs from the n-th sequence in the n-th position. According to the original assumption, any binary sequence (including the "antidiagonal" sequence just constructed) corresponds to a real number; however, none of the earlier assigned sequential numbers would correspond to our antidiagonal, which (obviously?) contradicts the assumption of the complete enumeration. This is to provide the substantial grounds for rejecting the hypothesis of the enumerability of real numbers, thus extending mathematics to however big "cardinal numbers" as the levels of actual infinity.

One could quite reasonably ask why, with several highly problematic assumptions for the premises of a deductive chain, the resulting contradiction should be interpreted to refute only one of them. Whence the immunity of the others? Our respect for an old tradition, or even earlier derivation of some statements as theorems in another theory, would in no way influence their vulnerability in another context, possibly inappropriate. With all the adherence to the axiomatic method, one can cancel any mathematical truth at all: a shadow of doubt about one of the axioms or the validity of the derivation scheme is quite enough. As it often happens, a seemingly innocent rearrangement, an extra brick, may bring down the whole construction. In the same way, a single contradiction is enough to ruin the whole of a mathematical enterprise rather than just a last minute extension.

For instance, take the convention that picking a certain digit from each of an infinite number of infinite binary sequences will produce an infinite sequence of the same kind. This is a direct reference to the well-(or ill-)known axiom of choice, whose admissibility still remains a cause of ardent controversy. To illustrate the nontrivial nature of the problem, let us simply reformulate Kantor's proof for transfinite numbers. Indeed, every natural number, just like real numbers, can be represented by a sequence of binary digits, with the only difference that, for integers, this sequence is "finite", that is, all its members are equal to zero after some sequence number (note that there is no effective procedure for establishing the fact of this finality in each particular case). Now, let as assume that all such sequences can be enumerated. Reproducing the same diagonal reasoning, we obtain a "number" that does not belong to the natural set. What's the difference? Compared to the original Kantor's deduction, here, we have originally enumerated only "finite" sequences, while the antidiagonal sequence is "infinite". It could still be considered as a representation of some integer number, which happens to be "infinitely big", and the existence of such numbers leads to a well-developed transfinite mathematics. Getting back to the school version, one is prompted to admit that, applied to some "countable" real numbers, Kantor's diagonal construction would produce a sequence representing a real number of a different kind, an "uncountable" real. This also gives way to a meaningful mathematical theory distinguishing the levels of "reality" by their inner complexity. Why not? We speak of transcendent numbers as different from mere irrational. More intricate distinctions might be useful in certain applications.

Of course, there are as legal conceptual alternatives. Let's fancy that the antidiagonal is similar to all the other binary sequences, and hence it must correspond to a traditional real number. With all that, this sequence is not entirely alien to any enumeration at all. Indeed, it can easily be included in the original enumeration: just increase all the original numbers by 1 and assign (ordinal) number 1 to the antidiagonal. Repeating the diagonal trick once again, we'll obtain yet another "extra" number, which still can be enumerated in the same manner. That is, real numbers cannot be enumerated once and forever, but there is no real number that would not be present in one of the possible dynamic enumerations. Real numbers are therefore both enumerable ant not enumerable. This contradiction indicates that, for infinite sequences, the very notion of enumeration needs better elaboration.

Just think about how the elements of a finite set could be enumerated. One would drive a hand into the sack, rummage an element out, and look for an (ordinal) number to attach. The procedure allows (at least) two modes of continuation. Thus, the exhaustion technique demand that we put the captive elements in a different bag (another set) and proceed with extracting the elements from the original set until it gets empty. In this approach an enumeration is represented by a sequence of non-intersecting pairs of sets, which is a very complex structure (often referred to as a partition of unity). An opposite choice is to feed the just enumerated element back to the original set and pick the next candidate from the same company. If the catch is already numbered, leave it as it is; otherwise, attach the next label and go on. The enumeration is then formally represented by a stochastic process, a probability distribution. In whatever case, there are certain objective complications. Sorting out the remainder is a poorly defined and slow-convergent procedure. In the first method, there is a problem of hunting the elements in extremely rarified media; the stochastic approach can never tell for sure if all the elements have been enumerated, or some statistically elusive individuals can still be present. When the elements of a set are from the material domain, they may sometimes be spoiled by the impact of enumeration. For example, if, in the exhaustion scheme, a catch of living fish is further kept on the ice (for the destination set), live numbers get gradually transformed into dead numbers; though the second method seems to be a little bit more humane, the numbered fish life time is not infinite in any way, and it may pass away before we are ready for the definitive verdict.

Well, the degree of animation of the object area may be of no importance in a particular theory. Still, this bring us back to the issue of essential qualities, which does not promise to make our problem any more tractable, since the theory is thus augmented with the necessity of formal derivation of the criteria of applicability.

With all that, we are inclined to think that the generalization of enumeration to infinite sets is a very complicated task hardly ever having a clear and unambiguous solution. Admit that somebody has invented a technology to enumerate the elements of a set in a quite definite manner, without recourse to random choice. That is, there is an effective procedure for computing the sequential number for all the numbers from a given class. Who can warrant that there no other numbers, of a different kind? Provided a new class has been discovered (or constructed), we need an appropriate enumeration scheme, as well as a rule for merging enumerations. Isn't is much simpler to admit that no infinity refers to any actual existence, meaning a currently developing process (possibly without end)? In this sense, the finite means the already completed. This is our present, while the past and the future belong to infinity.

To be honest, the situation is not that simple. The hierarchy of the present contains the appropriate levels to represent the past and the future. Similarly, in any infinity, there are finite structures, a kind of projection of the present, a shift of the reference point (compare the grammatical tenses like future in the past or future perfect). However, this matter is to be discussed elsewhere.

Get back to Kantor. We cannot tell beyond doubt, whether the lack of comprehension among the colleagues drove the poor guy to madness, or his innate insanity has ultimately infected the future scientists with crazy abstractions. Probably, the both ways. Still, it is absolutely clear: building a set theory for all in all, means building nothing at all. Unless we have agreed upon the nature of the entities that can become the elements of our sets, so that formal constructs would always imply an objective meaning, there is nothing to talk about, and no science but sheer meaninglessness. There is no reason for preferring one way to another, and no way to tell the right from the wrong. The objective perspective does not exclude the sets of abstract ideas; however, these ideas are to be naturally compatible, bundled by the inner logic of the object. To be sure, every object, beyond all the rest, exhibits the unity of opposite aspects, such as finality and infinity, discreteness and continuity, limits and unboundedness... Adapted to the mathematics of numbers, such universal, primary oppositions could logically be introduced in the theory from the very beginning, say, in the form of a natural sequence and a real axis. There is nothing to prove; this is the starting point for any further development. Armed with such a basic vision, one will get engaged in an honest investigation of the practically useful corollaries, or seek for a freaky bit, to come to a different object area, yet another aspect of reality. The attempts to formally substantiate what is determined by the objectively given order of things are nothing but a kind of ill soul-searching, warped reflection, a logical circularity and deductive self-reference, primitive diagonal reasoning. That is, a perversion and mental disease. Nevertheless, it's still for good, if crazy Kantor will lead the humanity to acquiring more respect for sound practical considerations.

Nov 1984