13.3 The Russell Paradox Many ordinary sets Other paradox options

The most famous of the already discovered in the XX century. of paradoxes is the antinomy discovered by B. Russell and described by him in a letter to G. Frege. The same antinomy was discussed simultaneously in Göttingen by German mathematicians 3. Zermelo and D. Hilbert.

The idea was in the air, and this appearance made an impression of a bomb exploded. This paradox has caused in mathematics, according to Gilbert, the effect of complete disaster. There is a threat over the most simple and important logical methods, the most common and useful concepts.

Immediately, the herd was obvious that neither in logic nor in mathematics in the entire long history of their existence had anything decidedly developed that could serve as a basis for eliminating antinomies. It was clearly necessary to move away from the usual ways of thinking. But from what place and in what direction? How radical was the rejection of established methods of theorizing?

With the further study of the antinomy, the conviction of the need for a fundamentally new approach grew steadily. Half a century after its discovery, specialists on the foundations of logic and mathematics L. Frenkel and I. Bar-Hillel asserted without any reservations: “We believe that any attempts to get out of the situation with the help of traditional ones (that is, those that were in use before the XX century) the ways of thinking, which have so far consistently failed, are obviously insufficient for this purpose. ”

Modern American logician X. Curry wrote a little later about this paradox: “In terms of logic, known in the 19th century, the situation simply could not be explained, although, of course, in our educated age there may be people who will see (or think what they will ) what is the mistake. ”

Russell's paradox in its original form is associated with the concept of a set, or class.

You can talk about the sets of different objects, for example, about the set of all people or about the set of natural numbers. Each separate person will be an element of the first set, each natural number will be an element of the second. It is also permissible to consider the sets themselves as some objects and to speak of sets of sets. You can even enter such concepts as the set of all sets or the set of all concepts.

Many ordinary sets

Regarding any arbitrary set, it seems meaningful to ask whether it is its own element or not. Sets that do not contain themselves as an element are called ordinary. For example, the set of all people is not a person, just as the set of atoms is not an atom. Unusual will be the sets that are own elements. For example, a set that unites all sets is a set and, therefore, contains itself as an element.

Consider now the set of all ordinary sets. Since it is many, one can also ask about it, whether it is ordinary or unusual. The answer, however, turns out to be discouraging. If it is ordinary, then, according to its definition, it must contain itself as an element, since it contains all the usual sets. But this means that it is an unusual set. The assumption that our set is the usual set, thus leads to a contradiction. So it cannot be ordinary. On the other hand, it cannot be unusual either: an unusual set contains itself as an element, and only ordinary sets are elements of our set. As a result, we come to the conclusion that the set of all ordinary sets cannot be ordinary, pi is an unusual set.

So, the set of all sets that are not proper elements is its own element if and only if it is not such an element. This is a clear contradiction. And it was obtained on the basis of the most plausible assumptions and with the help of seemingly incontestable steps.

Contradiction suggests that such a set simply does not exist. But why can't it exist? After all, it consists of objects that satisfy a well-defined condition, and the condition itself does not seem to be exceptional or unclear. If a set so simply and clearly cannot exist, then what, in fact, is the difference between possible and impossible sets? The conclusion about the non-existence of the set in question sounds unexpectedly and is disturbing. It makes our general notion of set amorphous and chaotic, and there is no guarantee that it is not capable of generating some new paradoxes.

The Russell Paradox is remarkable for its extreme commonality. To construct it, no complex technical concepts are needed, as in the case of some other paradoxes, the concepts of “set” and “element of set” are sufficient. But this simplicity speaks of its fundamentality: it touches the deepest foundations of our reasoning about sets, because it does not speak about any special cases, but about sets in general.

Other paradox options

The Russell Paradox is not specifically mathematical in nature. It uses the concept of a set, but does not affect any special properties related to mathematics. This becomes obvious if we reformulate the paradox in purely logical terms.

It is possible, in all likelihood, to ask about each property whether it applies to itself or not. The property of being hot, for example, is not applicable to oneself, since the property itself is not hot; the property of being concrete too ns refers to itself, for it is an abstract property. But the property of being abstract, being abstract, applies to itself. We call these properties inapplicable to themselves non- applicable. Does the property of being self-applicable apply? It turns out that irreducibility is not applicable only if it is not. This, of course, is paradoxical.

The logical, regarding the properties, variety of Russell's antinomy, is just as paradoxical as the mathematical, related to sets, its variety.

Russell also proposed the next popular version of the paradox he revealed.

Imagine that the council of one village defined the duties of a hairdresser in this way: to shave all the men of the village who do not shave themselves, and only these men. Should he shave himself? If so, he will treat those who shave himself, and those who shave himself should not shave. If not, he will belong to those who do not shave themselves, and that means he will have to shave himself. We thus arrive at the conclusion that this hairdresser shaves himself when and only if he does not shave himself. This is of course impossible.

The reasoning about the hairdresser rests on the assumption that such a hairdresser exists. The resulting contradiction means that this assumption is false, and there is no such inhabitant of the village who would shave all those and only those of its inhabitants who do not shave themselves.

The duties of a hairdresser do not seem at first glance contradictory, therefore the conclusion that it cannot be, sounds somewhat unexpected. But this conclusion is not all the same paradox. The condition that a village barber must satisfy is in fact internally contradictory and, therefore, impracticable. Such a hairdresser cannot be in a village for the same reason that there is no person in it who would be older than himself or who would have been born before his birth.

The reasoning about the hairdresser can be called a pseudoparadox. In its course, it is strictly analogous to the Russell paradox and this is interesting. But it is still not a true paradox.