Theorem and Axiom Lemma, Consequences

A theorem (ancient Greek θεώρημα - “proof, type; view; representation, position”) is a statement for which there is evidence in the theory under consideration (in other words, a conclusion). In contrast to theorems, axioms are called assertions that, within the framework of a specific theory, are accepted true without any evidence or justification.

In mathematical texts, theorems are usually called only those proven statements that are widely used in solving mathematical problems. At the same time, the required proofs are usually found by someone (the exception is mainly works on logic, in which the very concept of proof is studied, and therefore in some cases even indefinite statements are called theorems). Less important theorem statements are usually called lemmas, sentences, consequences, conditions, and other similar terms. Statements of which it is not known whether they are theorems are usually called hypotheses.

The most famous are the theorems of Fermat, Pythagoras, and Ptolemy.

The lemma (Greek: λημμα is an assumption) is a proven statement, useful not by itself, but to prove other statements. Examples of well-known lemmas are the Euclidean lemma, the Jordan lemma, the Gauss lemma, the Nakayama lemma, the Grindlinger lemma, the Lorentz lemma, and Lebema Lemma.

Axiom (ancient Greek ἀξίωμα - statement, position), postulate - the initial position of any theory that is accepted in the framework of this theory is true without the requirement of proof and is used in the basis of the proof of its other provisions. ^[one]

The need to accept axioms without evidence follows from inductive considerations: any evidence has to rely on any statements, and if for each of them to demand our own proofs, the chain will be infinite. In order not to go to infinity, it is necessary somewhere to break this chain - that is, to accept any statements without proof, as initial ones. Precisely such statements taken as the initial ones are called axioms. ^[2]

In modern science, axioms are those provisions of the theory that are taken as initial, and the truth question is solved either within the framework of other scientific theories or by interpreting this theory. ^[3]

Axiomatization theory is an explicit indication of a finite or countable, recursively enumerable (as, for example, in Peano axiomatics) a set of axioms and inference rules. After the names of the objects under study and their basic relations are given, as well as the axioms to which these relations must obey, all further presentation should be based solely on these axioms, not relying on the usual specific meaning of these objects and their relations. Axiom-based statements are called theorems .

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 it is impossible to arrive at a contradiction from the set of axioms, using the rules of logic, that is, to prove both a statement and its negation. Axioms are a kind of "starting points" for constructing theories in any science, while they themselves are not proved, but are derived directly from empirical observation (experience) or are substantiated in a deeper theory.

The Austrian mathematician Kurt Godel proved “incompleteness theorems” according to which any system of mathematical axioms (the formal system), starting from a certain level of complexity, is either internally contradictory or incomplete (that is, in fairly complex systems there will be at least one statement, the truth and falsity of which is not can be proved by means of this system itself). ^[four]

Examples of axioms

• Axiom of choice
• The axiom of parallelism of Euclid
• Archimedes axiom
• The axiom of bulk
• Axiom of regularity
• Axiom of complete induction
• Kolmogorov's axiom
• Boolean Axiom

• Axiomatics
  • Axiomatics of set theory
  • Axiomatics of real numbers
  • Euclidean axiomatics
  • Hilbert's axiomatics

Story

For the first time the term "axiom" is found in Aristotle (384-322 BC. E.) And moved to mathematics from the philosophers of ancient Greece. Euclid distinguishes between the concepts of "postulate" and "axiom", without explaining their differences. Since Boethius, the postulates are translated as requirements (petitio), axioms - as general concepts. Originally, the word "axiom" meant "truth, obvious in itself." In different manuscripts of Euclidean Beginnings, the division of statements into axioms and postulates is different, their order does not coincide. Probably the census takers held different views on the distinction between these concepts.

The attitude to axioms as to certain unchanging self-evident truths has been maintained for a long time. For example, in the Dahl dictionary, the axiom is “an evidence, clear in itself and an indisputable truth that does not require proof.”

Now axioms are not justified by themselves, but as necessary basic elements of the theory. Criteria for the formation of a set of axioms within a particular theory are often pragmatic: brevity of formulation, ease of manipulation, minimization of the number of initial concepts, etc. Such an approach does not guarantee the truth of accepted axioms. Only the confirmation of the theory is at the same time the confirmation of the set of its axioms. ^[one]

see also

• Dogma
• Concept
• Logics
• Hypothesis
• Formalism (mathematics)
• Gödel’s incompleteness theorems
• Reference system
• Fact
• Theorem
• Set theory
• Category Theory