The Russell Paradox presents two interdependent logical antinomies.
Two forms of Russell's paradox
The most frequently discussed form is a contradiction in the logic of sets. Some sets seem to be members of themselves, while others do not. The set of all sets is itself a set, so it seems that it refers to itself. Zero or empty, however, should not be a member of itself. Therefore, the set of all sets, as well as the zero, is not included in itself. The paradox arises when asked whether the multitude is a member of itself. This is possible if and only if it is not.
Another form of paradox is a contradiction regarding properties. Some properties seem to apply to themselves, while others do not. The property of being a property is itself 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 oneself? Again, the opposite follows from any assumption. The paradox was named after Bertrand Russell (1872–1970), who discovered 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 detail in his published works, the first to try to formulate solutions and the first to fully appreciate its significance. An entire chapter of the Principles was devoted to a discussion of this issue, and the appendix was devoted to the type theory that Russell proposed as a solution.
Russell discovered the "liar paradox" by examining the Cantor set theorem, which states that the power of any set is less than the set of its subsets. At least, there should be as many subsets in a domain 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 cannot be equal to the number of subsets. If there were the same number of them, then there would have to be a function ƒ that would map elements to their subsets. At the same time, it can be proved that this is impossible. Some elements can be mapped by the ƒ function to the subsets that contain them, while others cannot.
Consider the subset of elements that do not belong to their images into which ƒ maps them. 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 onto which ƒ maps it. This is only possible if it does not belong. Russell's paradox can be seen as an example of the same line of reasoning, only simplified. What more - sets or subsets of sets? It would seem that there should be more sets, since all subsets of the sets themselves are sets. But if Cantor's theorem is true, then more subsets must exist. Russell considered the simplest mapping of sets onto themselves and applied the Cantorian approach of considering the set of all these elements that are not included in the sets into which they are mapped. The Russell map becomes the set of all sets that are not included in itself.
Frege Error
The Liar Paradox has had profound implications 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 each condition or predicate to be defined, we can assume that there are only a lot of things that satisfy this condition. The variant of the paradox concerning properties - a natural continuation of the version with sets - raised serious doubts as to whether it is possible to assert the objective existence of a property or universal correspondence to each defined condition or predicate.
Soon, contradictions and problems were found in the work of those logicians, philosophers and mathematicians who made such assumptions. In 1902, Russell discovered that a version of the paradox can be expressed in the logical system developed in the first volume of Gottlob Frege's “Foundations of Arithmetic”, one of the main works on the logic of the late XIX - early XX centuries. In Frege's philosophy, many are understood as “expansion” 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 its defining concept only if it is not.
Russell wrote to Frege about this contradiction in June 1902. Correspondence has become 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 regarding properties was resolved in his philosophy by distinguishing between levels of concepts.
Frege understood concepts as functions of the transition from arguments to truth values. The concepts of the first level take objects as arguments, the concepts of the second level take these functions as arguments and so on. Thus, a concept can never take itself as an argument, and a paradox regarding properties cannot be formulated. Nevertheless, Frege understood sets, extensions, or concepts as belonging to the same logical type as all other objects. Then for each set the question arises whether it falls under the concept that defines it.
When Frege received Russell’s first letter, the second volume of the Foundations of Arithmetic was already finished. He was forced to quickly prepare an application that responded to Russell's paradox. Frege’s examples contained a number of possible solutions. But he came to the conclusion that weakened the concept of abstraction of a set in a logical system.
In the original, it was possible to conclude that an object belongs to the set if and only if it falls within the concept that defines it. In the revised system, we can only conclude that an object belongs to a set if and only if it falls under the concept of a defining set, and not the set in question. The Russell paradox does not arise.
The decision, however, did not entirely satisfy Frege. And that was the reason. A few years later, a more complex form of contradiction was found for the revised system. But even before this happened, Frege rejected his decision and seems to have come to the conclusion that his approach was simply inoperative and that 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 response to the paradoxes of set theory in a variant formulated for properties. Frege's answer preceded the most frequently discussed solution to this form of paradox. It is based on the fact that properties fall under different types and that the type of property is never the same as the elements to which it refers.
Thus, the question does not even arise whether the property is applicable to itself. A logical language that separates elements according to such a hierarchy uses type theory. Although it is already used by Frege, for the first time Russell fully explained and justified it in the Appendix to the Principles. Type theory was more complete than distinguishing Frege levels. She divided properties not only into various logical types, but also sets. Type theory resolved the contradiction in Russell's paradox as follows.
In order to be philosophically adequate, the adoption of type theory for properties requires the development of a theory about the nature of properties in such a way as to explain why they cannot be applied to themselves. At first glance, it makes sense to predict your own property. The property of being self-identical, it would seem, is also self-identical. The property of being pleasant seems pleasing. Similarly, it seems to be 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 its part, the rationale for Frege’s separation of different levels of concepts comes from his theory of unsaturation of concepts. Concepts as functions are essentially incomplete. To provide a value, they need an argument. One cannot simply predict one concept with a concept of the same type, since it still requires its own argument. For example, while 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 get the result.
About 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 as a whole, then, if we accept nominalism, the paradox can be completely avoided.
However, to solve the antinomy one does not need to be so extreme. The higher-order logical systems developed by Frege and Russell contained, as they say, a conceptual principle according to which for every open formula, no matter how complex it is, there is a property or concept as an element, using only those things that satisfy the formula as an example. They applied to 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, giving the right of objective existence to simple properties, including, for example, red, hardness, kindness, etc. You could even allow these properties to apply to themselves, for example, kindness can to be kind.
And the same status for complex attributes can be denied, for example, for such “properties” as having-seventeen-heads, being-written-underwater, etc. In this case, no specified condition corresponds to a property understood as separately 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 itself and avoiding the paradox by applying a more conservative metaphysics of properties.
The Russell 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 to the 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 basic information about which is presented below.
Type Theory for Sets
As mentioned earlier, Russell advocated a more complete type theory, which would divide not only properties or concepts into different types, but also sets. Russell divided sets into sets of separate objects, sets of sets of separate objects, etc. Sets were not considered objects, and sets of sets were sets. Many have never possessed a type that allows them to have themselves as members. Therefore, there is no set of all sets that are not members of their own, because for any set the question of whether it is its member is 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, V.V. Quine proposed an alternative solution, somewhat similar to type theory. The basic information about him is as follows.
Separation by an element, sets, etc., is carried out in such a way that the assumption that the set is in itself is always incorrect or meaningless. Many can exist only under the condition that the conditions determining them are not a type violation. Thus, for Quine, the expression “x is not a member of x” is a significant statement that does not imply the existence of the set of all elements of x satisfying this condition.
In this system, a 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 sign of occurrence in the set, the variable assigned to it is assigned one less than the variable, next after him. This blocks the Russell paradox, because the formula used to determine the problem set has the same variable before and after the membership sign, which makes it non-stratified.
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 is adopted in the theory of Zermelo-Frenkel sets (TF). Here, the restriction on the existence of sets is also established. Instead of Russell and Frege's “top-down” approach, which initially believed that for any concept, property or condition, we can assume the existence of a multitude of all things with such a property or satisfying such a condition, everything begins in the CF theory “from the bottom up”.
The individual elements and the empty set form the set. Therefore, in contrast to the early Russell and Frege systems, the FF does not belong to the universal set, which includes all elements and even all sets. The TF sets strict limits on the existence of sets. Only those can exist for which it is explicitly postulated or which can be compiled using iterative processes, etc.
Then, instead of the notion of abstraction of a naive set, which states that an element is included in a certain set if and only if it meets the defining condition, the principle of separation, selection, or “sorting” is used in the CF. 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 elements in the original set that satisfies the condition.
Then the principle of abstraction comes in: if the set A exists, then for all elements x in A, x belongs to the subset A that satisfies condition C if and only if x satisfies condition C. This approach solves the Russell paradox, since we cannot just assume that there are many of all sets that are not members of themselves.
Having many sets, you can select 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 the Russell problem set, the contradiction cannot be proved.
Other solutions
In addition, there have been subsequent extensions or modifications to all of these solutions, such as the branching of the type theory of the Principles of Mathematics, the extension of Quine's System of Mathematical Logic, and later developments in set theory by Bernays, Godel, and von Neumann. The question of whether an answer has been found to the insoluble paradox of Bertrand Russell is still the subject of debate.