Аксиоматика тарского (1920-тые – 1980-тые) Fundamental relations

These axioms are a more elegant version of a set Tarski devised in the 1920s as part of his investigation of the metamathematical properties of Euclidean plane geometry. This objective required reformulating that geometry as a first-order theory.

Tarski did so by positing a universe of points, with lower case letters denoting variables ranging over that universe. 

Equality is provided by the underlying logic.

 Tarski then posited two primitive relations:

• Betweenness, a triadic relation. The atomic sentence Bxyz denotes that y is "between" x and z, in other words, that y is a point on the line segment xz. (This relation is interpreted inclusively, so that Bxyz is trivially true whenever x=y or y=z).

• Congruence (or "equidistance"), a tetradic relation. The atomic sentence wx ≡ yz can be interpreted as wx is congruent to yz, in other words, that the length of the line segment wx is equal to the length of the line segment yz.

Betweenness captures the affine aspect of Euclidean geometry; congruence, its metric aspect. The background logic includes identity, a binary relation. The axioms invoke identity (or its negation) on five occasions.

The axioms below are grouped by the types of relation they invoke, then sorted, first by the number of existential quantifiers, then by the number of atomic sentences. The axioms should be read as universal closures; hence any free variables should be taken as tacitly universally quantified.

10 Axioms and one axiom schema shown below Congruence axioms

Reflexivity of Congruence

The distance from x to y is the same as that from y to x. This axiom asserts a property very similar to symmetry for binary relations.

Identity of Congruence

If xy is congruent with a segment that begins and ends at the same point, x and y are the same point. This is closely related to the notion of reflexivity for binary relations.

Transitivity of Congruence

Two line segments both congruent to a third segment are congruent to each other; all three segments have the same length. This axiom asserts that congruence is Euclidean, in that it respects the first of Euclid's "common notions." Hence this axiom could have been named "Congruence is Euclidean." The transitivity of congruence is an easy consequence of this axiom and Reflexivity.

Betweenness axioms

Identity of Betweenness

The only point on the line segment  is  itself.

Pasch's axiom

Axiom of Pasch

Draw line segments connecting any two vertices of a given triangle with the sides opposite the vertices. These two line segments must then intersect at some point inside the triangle.

Axiom schema of Continuity

Continuity: φ and ψ divide the ray into two halves and the axiom asserts the existence of a point b dividing those two halves

Let φ(x) and ψ(y) be first-order formulae containing no free instances of either a or b. Let there also be no free instances of x in ψ(y) or of y in φ(x). Then all instances of the following schema are axioms:

Let r be a ray with endpoint a. Let the first order formulae φ and ψ define subsets X and Y of r, such that every point in Y is to the right of every point of X (with respect to a). Then there exists a point b in r lying between X and Y. This is essentially the Dedekind cut construction, carried out in a way that avoids quantification over sets.