Axiom

see also: Etymology of Axiom

The word axiom comes from the Greek αξιωμα ( axioma ), which means " who is regarded as worthy or convenable" or " who is regarded as obvious in soi" . For some Philosophe S Greek of antiquity that represented an assertion which they regarded as obvious and which did not have no need for proof. The word comes from αξιοειν ( axioein ), meaning to regard as worthy, itself derived from αξιος ( axios ), meaning worthy.

In epistemology, an axiom is an obvious truth in oneself on which another knowledge can rest, in other words can be built above. Let us specify that all the epistemologists do not admit that the axioms, in this direction of the term, exist. In certain philosophical currents, like the Objectivism, the word “axiom” has a particular connotation. A statement is axiomatic if it is impossible to deny it without being contradicted. Example: “There exists an absolute truth” or “the language exists” are axioms.

In Mathématiques the word axiom indicated a proposal which is obvious in oneself in the Greek mathematical tradition, as in the Éléments of Euclide. The axiom is used from now on, in Logique mathematics, to indicate a truth first, inside a Théorie. The whole of the axioms of a Théorie is called axiomatic . This axiomatic must of course be not-contradictory; it is its only constraint. This axiomatic defines the theory; what means that the axiom cannot be called into question inside this theory, it is said whereas this theory is consistent. An axiom thus represents rather a starting point in a system of logic and it can be arbitrarily selected. Of course, the relevance of a theory depends on the relevance of its axioms and its interpretation. Actually, it is noncoherence of its interpretation, which the refutation comes from the not-contradictory theory, and consequently, of its axiomatic. The axiom is thus with mathematical logic, which is the Postulat with the Theoretical physics. rings, the operation of the multiplication is commutative, and in others it is not it; these rings in which the law is commutative satisfy the “axiom of the commutation of the multiplication”. One confused axiom and Postulat a long time, although one already differentiates them in the Éléments from Euclide (the axioms are obvious whereas one asks to admit the postulates). --> Axioms are used as a basis elementary for any system of formal Logique. For example, one can define arithmetic simple, including/understanding an addition, while posing (by being inspired a little Peano):

a number noted 0 exists
any number X has a successor noted succ (X)
X+0 = X
succ (X) + Y = X + succ (Y)

Many Théorème S can be shown starting from these axioms.

By using these axioms, and by defining the usual words 1,2,3, and so on to designate the successors of 0 succ (0), succ (succ (0)), succ (succ (succ (0))) respectively, we can show what follows:

succ (X) = X + 1

and

1 + 2 = 1 + succ (1) Expansion of the abbreviation (2 = succ (1))

1 + 2 = succ (1) + 1 Axiom 4

1 + 2 = 2 + 1 Expansion of the abbreviation (2 = succ (1))

1 + 2 = 2 + succ (0) Expansion of the abbreviation (1 = succ (0))

1 + 2 = succ (2) + 0 Axiom 4

1 + 2 = 3 Axiom 3 and use of the abbreviation (succ (2) = 3)

Any result that we can deduce from the axioms does not need to be an axiom. Any assertion which cannot be deduced from the axioms and whose negation cannot result either from these same axioms, can reasonably be added like axiom.

Probably oldest and also the most famous system of axioms is that of the 4+1 postulates of Euclide. Those prove to currently be rather incomplete, and much more postulates are necessary to characterize the geometry of Euclide completely (Hilbert used of them 26 in its axiomatic of the Euclidean geometry).

Each one of these choices gives us various alternative forms of geometry, in which measurements of the interior angles of a triangle are added to give a lower value, equal or higher than the measurement of the angle formed by a line (flat angle). These geometries are known as elliptic geometries , Euclidean and hyperbolic respectively. the general theory of relativity is based primarily on an assertion which the mass gives to space a curve.

The fact that alternative forms of geometry could exist, worried much the mathematicians of the 19th century and in similar developments, for example in Boolean algebra, they generally made efforts to deduce the results from the systems from arithmetic ordinary. Welsh showed that all these efforts were mainly wasted, and that the parallel developments of the axiomatic systems could be used advisedly, in the same manner that it algebraically solved many problems of traditional geometry.

Finally, the abstract similarities existing between the algebraic systems were perceived like more important than the details and the modern algebra had been born.

At the 20th century, the theorem of incomplétude of Gödel proves that no list clarifies axioms sufficient to deduce the principle from recurrence on the entireties could not be at the same time complete (each proposal can be shown or refuted inside the system) and consistent (no proposal can be at the same time shown and refuted).

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