 applications.
The extraordinary expansion of the subject of mathematics in the 19th century resulted in increased attention to the problems of the “substantiation” of mathematics, that is, to a critical reevaluation of initial assumptions (axioms), to the construction of a rigorous system of definitions and proofs, and to a critical examination of the logical means used in these proofs. Works on the rigorous substantiation of various branches of mathematics rightly occupied a significant place in 19th- and 20th-century mathematics. The results obtained in the substantiation of the foundations of analysis (the theory of real numbers, the theory of limits, and the rigorous substantiation of all methods of the differential and integral calculus) are now given with greater or lesser completeness in most textbooks (even purely practical textbooks). However, there are still instances where the rigorous substantiation of a mathematical theory arising from practical requirements is delayed. As late as the turn of the 20th century this was the case with operational calculus, which found extremely broad applications in mechanics and electrical engineering. A logically irreproachable exposition of mathematical probability theory was given only after a long delay. And even now there is no rigorous substantiation of many mathematical methods that are used extensively in modern theoretical physics, where many valuable results are obtained by “illegal” mathematical procedures.
The standards of mathematical rigor currently observed by practicing mathematicians took shape only toward the end of the 19th century. These standards are based on the set-theoretic conception of the structure of any mathematical theory. From this point of view any mathematical theory deals with one or several sets of objects that are interconnected by certain relations. All formal properties of these objects and relations necessary for the development of the theory are stated in the form of axioms that do not touch upon the specific nature of the objects and relations themselves. A theory is applicable to any system of objects connected by relations that satisfy its fundamental system of axioms. Accordingly, a theory is considered to be logically rigorous if it employs only those properties of the objects or of the relations that are mentioned in the axioms and if all additional objects or relations that are introduced in the course of the development of the theory are defined formally in terms of the axioms.
Mathematical logic clarifies another aspect of the structure of any mathematical theory. A system of axioms in the (set-theoretic) sense outlined above now merely imposes “outer” limits on the domain of applications of a given mathematical theory. It indicates the properties of the system of objects and relations under study but provides no clues concerning the logical means by which this mathematical theory may be developed. For example, the properties of the system of natural numbers can be determined up to the accuracy of isomorphism by using a very simple system of axioms. Nevertheless, it often proves to be difficult to solve problems the answer to which in principle is uniquely predetermined by the adoption of this system of axioms: number theory abounds in problems that were posed and very simply formulated long ago but that have not yet been solved. The question naturally arises whether this is so because the solution of a few simply formulated problems of number theory requires a very long chain of arguments composed of elementary links that are known and already have come into use, or because essentially new, not previously used, methods of logical derivation are required to solve certain problems of number theory.
Modern mathematical logic has provided a specific answer to this question: no single deductive theory can exhaust the diversity of the problems of number theory. More accurately, even within the theory of natural numbers it is possible to formulate a sequence of problems p1,p2, …, pn, … such that for any deductive theory there will be