Axiom

The need to adopt axioms without proof follows from an inductive consideration: any proof is forced to rely on some statements, and if each of them requires its own proofs, the chain will be infinite. In order not to go to infinity, you need to break this chain somewhere - that is, to accept some statements without proof, as the initial ones. It is precisely such statements, accepted as initial ones, that are called axioms ^[2] .

In modern science, the question of the truth of the axioms underlying any theory is solved either in the framework of other scientific theories, or through the interpretation of this theory ^[3] .

Axiomatization of a theory is an explicit indication of a finite or countable , recursively enumerable (as, for example, in Peano's axiomatics ) set of axioms and inference rules. After the names are given to the studied objects and their basic relations, as well as the axioms to which these relations must obey, all further exposition should be based solely on these axioms and not be based on the usual concrete meaning of these objects and their relations.

The choice of axioms that form the basis of a particular theory is not the only one. Examples of different but equivalent sets of axioms can be found in mathematical logic and Euclidean geometry .

A set of axioms is called consistent if, based on the axioms of the set, using the rules of logic, it is impossible to come to a contradiction, that is, to prove at the same time a certain statement and its negation .

Austrian mathematician Kurt Gödel proved the " incompleteness theorems ", according to which any system of mathematical axioms ( formal system ) in which natural numbers, addition and multiplication can be defined is incomplete. This means that there is an infinite number of mathematical statements (functions, expressions), neither the truth nor the falsity of which can be proved on the basis of this system of axioms. Also, by the incompleteness theorem, among these irreducible statements there will be a statement about the consistency of this system.

For the first time the term “axiom” is found in Aristotle ( 384 - 322 BC ) and goes into mathematics from the philosophers of Ancient Greece . Euclid distinguishes between the concepts of "postulate" and "axiom", without explaining their differences. Since Boethius, postulates have been translated as requirements (petitio), axioms as general concepts. Initially, the word "axiom" had the meaning of "truth, obvious in itself." In different manuscripts of the “Beginnings” of Euclid, the division of statements into axioms and postulates is different, their order does not coincide. Probably, scribes held different views on the difference between these concepts.

The attitude to axioms as to certain invariable self-evident truths remained for a long time. For example, in Dahl’s dictionary, the axiom is “evidence, clear in itself and indisputable truth , requiring no evidence ”.

The impetus for the change in the perception of axioms was the work of the Russian mathematician Nikolai Lobachevsky on non-Euclidean geometry , first published in the late 1820s. While still a student, he tried to prove the fifth postulate of Euclid, but later abandoned this. Lobachevsky concluded that the fifth postulate is only an arbitrary restriction, which can be replaced by another restriction. If the fifth postulate of Euclid were provable, then Lobachevsky would encounter contradictions. However, although the new version of the fifth postulate was not clearly obvious, it fully served as an axiom, making it possible to construct a new consistent geometry system.

At first, Lobachevsky’s ideas were not recognized (for example, academician Ostrogradsky spoke negatively of them). Later, when Lobachevsky published his work in other languages, he was noticed by Gauss , who also had some groundwork in the field of non-Euclidean geometry. He indirectly expressed admiration for this work. Lobachevsky ’s geometry received real recognition only 10-12 years after the author’s death, when its consistency was proved in the case of Euclidean geometry consistency. This led to a revolution in the mathematical world. Hilbert launched a large-scale project on the axiomatization of all mathematics to prove its consistency. His plans were not destined to come true because of Gödel's subsequent incompleteness theorems . However, this served as an impetus for the formalization of mathematics. For example, the axioms of natural numbers and their arithmetic appeared , as well as Cantor’s work on the creation of set theory . This allowed mathematicians to create strictly true proofs for theorems.

Now the axioms are not justified on their own, but as necessary basic elements of the theory - axioms can be quite arbitrary, they do not have to be obvious. The only invariable requirement for axiomatic systems is their internal consistency. The criteria for the formation of a set of axioms in the framework of a particular theory are often pragmatic: brevity of the wording, ease of manipulation, minimization of the number of initial concepts, etc. Such an approach does not guarantee the truth of the accepted axioms ^[1] . According to Popper’s criterion , the only negative example refutes the theory and, as a result, proves the falsity of the system of axioms, while a lot of supporting examples only increase the probability of the truth of the system of axioms.

Examples of axioms

1. Axiom of choice
2. Axiom of Euclidean parallelism
3. Axiom of Archimedes
4. Axiom of volume
5. Axiom of regularity
6. The axiom of complete induction
7. Axiom Kolmogorov
8. Axiom of Boolean .

Examples of axiom systems

1. Axiomatics of set theory
2. Axiomatics of real numbers
3. Axiomatics of Euclid
4. Hilbert axiomatics .

• Dogma
• Common sense
• Concept
• Logics
• Hypothesis
• Formalism (Mathematics)
• Gödel's incompleteness theorem
• Reference system
• Fact
• Theorem
• Set theory
• Category theory

• The beginning of Euclid. Books I — VI. M.-L., 1950
• Hilbert D. Foundations of geometry. M.-L., 1948

• Axiom // Brockhaus and Efron Encyclopedic Dictionary : in 86 volumes (82 volumes and 4 additional). - SPb. , 1890-1907.