The Russell Paradox: background, examples, wording

The Russell paradox represents two interdependent logical antinomies.

Two Forms of the Russell Paradox

The most frequently discussed form is the contradiction in the logic of sets. Some sets, it seems, can be members of themselves, and others - not. The set of all sets is itself a set, therefore it seems that it refers to itself. Zero or empty, however, should not be a member of yourself. Therefore, the set of all sets, like zero, does not enter into itself. The paradox arises at the question of whether the set is a member of itself. This is possible if and only if it is not so.

Another form of paradox is a contradiction concerning properties. Some properties seem to belong to themselves, while others do not. The property of being a property in itself is a property, while the property of being a cat is not. Consider the property of having a property that does not apply to itself. Is it applicable to itself? Again, from any assumption follows the opposite. The paradox was named after Bertrand Russell (1872-1970), who opened it in 1901.

History

Russell's discovery occurred during his work on the "Principles of Mathematics." Although he discovered the paradox on his own, there is evidence that other mathematicians and developers of set theory, including Ernst Zermelo and David Hilbert, knew about the first version of the contradiction before him. Russell, however, was the first to discuss the paradox in his published works, first he tried to formulate solutions and was the first to fully appreciate its significance. An entire chapter of the "Principles" was devoted to a discussion of this issue, and the annex was devoted to the theory of types, which Russell proposed as a solution.

Russell discovered the "liar paradox", considering the theorem of Cantor sets that the power of any set is less than the set of its subsets. At least in a domain there should be as many subsets as there are elements in it, if for each element one subset is a set containing only this element. In addition, Cantor proved that the number of elements can not be equal to the number of subsets. If they had the same number, then there would have to be a function ƒ that would map the elements to their subsets. At the same time, it can be proved that this is impossible. Some elements can be displayed by the function ƒ on subsets that contain them, while others can not.

Consider a subset of elements that do not belong to their images, into which they are mapped. It itself is a subset of the elements, and therefore the function ƒ would have to map it to some element in the domain. The problem is that then the question arises as to whether this element belongs to the subset to which it is mapped. This is possible only if it does not belong. The Russell paradox can be considered as an example of the same line of reasoning, only simplified. What are more - sets or subsets of sets? It would seem that there should be more sets, since all subsets of sets themselves are sets. But if the Cantor theorem is true, then there must be more subsets. Russell considered the simplest mapping of sets onto themselves and applied the Cantorian approach to considering the set of all these elements that do not belong to the sets into which they are mapped. The Russell map becomes the set of all sets that do not belong to itself.

Error Frege

"The paradox of a liar" had profound consequences for the historical development of set theory. He showed that the concept of a universal set is extremely problematic. He also questioned the notion that for every defined condition or predicate we can assume the existence of a set of only those things that satisfy this condition. The variant of the paradox concerning properties - the natural continuation of the version with sets - raised serious doubts as to whether it is possible to assert about the objective existence of a property or universal correspondence to each determined condition or predicate.

Soon, contradictions and problems were found in the works of those logicians, philosophers and mathematicians who made similar assumptions. In 1902, Russell discovered that the paradox can be expressed in the logical system developed in Volume I of Gottlob Frege, The Foundations of Arithmetic, one of the main works on the logic of the end of the nineteenth and early twentieth centuries. In Frege's philosophy, a set is understood as an "extension" or "value-range" of a concept. Concepts are the closest correlates to properties. It is assumed that they exist for each given state or predicate. Thus, there is a concept of a set that does not fall under its defining concept. There is also a class defined by this concept, and it falls under the defining concept only if it is not so.

Russell wrote to Frege about this contradiction in June 1902. Correspondence became one of the most interesting and discussed in the history of logic. Frege immediately recognized the catastrophic consequences of the paradox. He noted, however, that the version of the contradiction concerning properties in his philosophy was resolved by distinguishing the levels of concepts.

The concept of Frege was understood as a function of the transition from arguments to truth values. The concepts of the first level accept objects as arguments, second-level concepts take these functions as arguments and so on. Thus, the concept can never take itself as an argument, and the paradox regarding properties can not be formulated. Nevertheless, sets, extensions or concepts were understood by Frege as belonging to the same logical type as all other objects. Then for each set there is a question, whether it falls under the concept defining it.

When Frege received Russell's first letter, the second volume of "Foundations of Arithmetic" was already finishing. He was forced to quickly prepare an application that would answer the Russell paradox. Frege's examples contained a number of possible solutions. But he came to a conclusion that weakened the concept of the abstraction of the set in the logical system.

In the original it was possible to come to the conclusion that an object belongs to a set if and only if it falls under the concept determining it. In a revised system, one can only conclude that an object belongs to a set if and only if it falls under the notion of a defining set, and not the set in question. Russell's paradox does not arise.

The decision, however, did not quite satisfy Frege. And that was the reason. A few years later, for a revised system, a more complex form of contradiction was found. But even before this happened, Frege refused his decision and, it seems, came to the conclusion that his approach was simply inefficient, and that the logicians would have to do without sets at all.

Nevertheless, other, relatively more successful alternative solutions were proposed. They are discussed below.

Type Theory

It was noted above that Frege had an adequate answer to the paradoxes of set theory in the variant formulated for properties. Frege's answer preceded the most frequently discussed solution of this form of paradox. It is based on the fact that properties fall into different types and that the type of property is never the same as the elements to which it relates.

Thus, the question does not even arise whether the property is applicable to itself. A logical language that separates elements from such a hierarchy uses type theory. Although it is already used by Frege, for the first time it was fully explained and justified by Russell in the Appendix to the Principles. The theory of types was more complete than the distinction between Frege levels. It shared properties not only into different logical types, but also sets. The theory of types solved the contradiction in the Russell paradox as follows.

In order to be philosophically adequate, the acceptance of type theory for properties requires the development of a theory about the nature of properties in such a way that one can explain why they can not be applied to themselves. At first glance, it makes sense to predicate your own property. The property of being self-identical, it would seem, is also self-identical. The property of being pleasant seems pleasant. Similarly, apparently, it seems false to say that the property of being a cat is a cat.

Nevertheless, various thinkers justified the division of types in different ways. Russell even gave different explanations at different times of his career. For his part, the substantiation of Frege's division of different levels of concepts proceeds from his theory of unsaturation of concepts. Concepts, as functions, are essentially incomplete. To supply a value, they need an argument. One can not simply predicate one concept by a concept of the same type, because it still requires its argument. For example, although it is still possible to extract the square root from the square root of a certain number, it is not possible to simply apply the square root function to the square root function and obtain the result.

On the conservatism of properties

Another possible solution to the property paradox is to deny the existence of a property in accordance with any given conditions or a well-formed predicate. Of course, if someone eschews metaphysical properties as objective and independent elements in general, then, if nominalism is accepted, the paradox can be completely avoided.

However, to solve the antinomy need not be so extreme. The higher-order logical systems developed by Frege and Russell contained, as they say, the conceptual principle that for every open formula, no matter how complex it is, there exists as an element a property or a concept in the example of only those things that satisfy the formula. They were applied to the attributes of any possible set of conditions or predicates, no matter how complex they were.

Nevertheless, one could adopt a more rigorous metaphysics of properties, granting the right of objective existence to simple properties, including, for example, red color, hardness, kindness, etc. One can even allow these properties to be applied to themselves, for example, kindness can Be kind.

And the same status for complex attributes can be denied, for example, for such "properties" as having-seventeen heads, being-written-under-water, etc. In this case, no given condition corresponds to a property understood as a separate An existing element that has its own properties. Thus, one can deny the existence of a simple property of being-property-which-is-not-applicable-to-self and avoiding the paradox by applying a more conservative metaphysics of properties.

Russell's paradox: the solution

It was noted above that at the end of his life Frege completely abandoned the logic of sets. This, of course, is one solution of antinomy in the form of sets: a simple denial of the existence of such elements as a whole. In addition, there are other popular solutions, the main details of which are presented below.

Theory of types for sets

As mentioned earlier, Russell advocated a more complete theory of types that would separate not only properties or concepts into different types, but also sets. Russell divided the sets into sets of individual objects, sets of sets of individual objects, etc. Sets were not considered objects, and sets of sets were sets. The set never had a type that allows itself to have itself as a member. Therefore, there is no set of all sets that are not proper terms, because for any set the question of whether it is a member is in itself a type violation. Again, the problem here is to clarify the metaphysics of sets in order to explain the philosophical foundations of division into types.

Stratification

In 1937, VV Quine proposed an alternative solution, in some ways similar to the theory of types. The basic information about him is as follows.

Separation by an element, sets, etc. is done in such a way that the assumption of finding the set in itself is always wrong or meaningless. Sets can exist only under the condition that the conditions defining them are not a violation of types. Thus, for Quine, the expression "x is not a member of x" is an important statement that does not imply the existence of the set of all elements x that satisfy this condition.

In this system, the set exists for some open formula A if and only if it is stratified, that is, if the variables are assigned natural numbers in such a way that for each attribute of the occurrence in the set of the preceding variable, the assignment is assigned one less than the variable, Next to it. This blocks the Russell paradox, since the formula used to determine the problem set has the same variable before and after the membership sign, which makes it unstratified.

However, it remains to be determined whether the resulting system, which Quine called "New Foundations of Mathematical Logic," is consistent.

Sorting

A completely different approach has been adopted in Zermelo-Fraenkel set theory (ZF). Here, too, a restriction on the existence of sets is established. Instead of the "top-down" approach of Russell and Frege, who initially believed that for any concept, property or condition, it is possible to assume the existence of a set of all things with such a property or satisfying such a condition, in the CF theory everything starts "from the bottom up".

The individual elements and the empty set form a set. Therefore, unlike the early systems of Russell and Frege, the FT does not belong to the universal set, which includes all elements and even all sets. The FT sets rigid restrictions on the existence of sets. There can only exist those for which it is explicitly postulated or that can be compiled using iterative processes, and so on.

Then, instead of the notion of the abstraction of a naive set, which says that an element is included in a certain set if and only if it meets a defining condition, the principle of separation, separation or "sorting" is used in the FT. Instead of assuming the existence of a set of all elements that, without exception, satisfy a certain condition, for each already existing set, sorting indicates the existence of a subset of all the elements in the original set that satisfies the condition.

Then the abstraction principle comes: if the set A exists, then for all elements x in A, x belongs to the subset A that satisfies the condition C if and only if x satisfies condition C. This approach solves the Russell paradox, since we can not simply assume That there is a set of all sets that are not members of themselves.

Having a set of sets, we can distinguish or divide it into sets that are in themselves, and those that are not, but since there is no universal set, we are not connected by the set of all sets. Without the assumption of Russell's problem set, a contradiction can not be proved.

Other solutions

In addition, subsequent extensions or modifications of all these solutions took place, such as the ramification of the theory of types of "Principles of mathematics", the expansion of the system of "Mathematical Logic" by Quine, as well as the later developments in set theory made by Bernays, Gödel and von Neumann. The question of whether the answer to the intractable paradox of Bertrand Russell is found is still a matter of debate.