Tarski's axioms

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 (see First-order logic#Equality and its axioms).[1] 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.

Reflexivity of Congruence

${\displaystyle xy\equiv yx\,.}$

Identity of Congruence

${\displaystyle xy\equiv zz\rightarrow x=y.}$

Transitivity of Congruence

${\displaystyle (xy\equiv zu\land xy\equiv vw)\rightarrow zu\equiv vw.}$

While the congruence relation ${\displaystyle xy\equiv zw}$ is, formally, a 4-way relation among points, it may also be thought of, informally, as a binary relation between two line segments ${\displaystyle xy}$ and ${\displaystyle zw}$. The "Reflexivity" and "Transitivity" axioms above, combined, prove both:

• that this binary relation is in fact an equivalence relation
  • it is reflexive: ${\displaystyle xy\equiv xy}$.
  • it is symmetric ${\displaystyle xy\equiv zw\rightarrow zw\equiv xy}$.
  • it is transitive ${\displaystyle (xy\equiv zu\land zu\equiv vw)\rightarrow xy\equiv vw}$.
• and that the order in which the points of a line segment are specified is irrelevant.
  • ${\displaystyle xy\equiv zw\rightarrow xy\equiv wz}$.
  • ${\displaystyle xy\equiv zw\rightarrow yx\equiv zw}$.
  • ${\displaystyle xy\equiv zw\rightarrow yx\equiv wz}$.

The "transitivity" axiom asserts that congruence is Euclidean, in that it respects the first of Euclid's "common notions".

The "Identity of Congruence" axiom states, intuitively, that 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.

Pasch's axiom

Identity of Betweenness

${\displaystyle Bxyx\rightarrow x=y.}$

The only point on the line segment ${\displaystyle xx}$ is ${\displaystyle x}$ itself.

Axiom of Pasch

${\displaystyle (Bxuz\land Byvz)\rightarrow \exists a\,(Buay\land Bvax).}$

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

Axiom schema of Continuity

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:

${\displaystyle \exists a\,\forall x\,\forall y\,[(\phi (x)\land \psi (y))\rightarrow Baxy]\rightarrow \exists b\,\forall x\,\forall y\,[(\phi (x)\land \psi (y))\rightarrow Bxby].}$

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.

Lower Dimension

${\displaystyle \exists a\,\exists b\,\exists c\,[\neg Babc\land \neg Bbca\land \neg Bcab].}$

There exist three noncollinear points. Without this axiom, the theory could be modeled by the one-dimensional real line, a single point, or even the empty set.

Upper dimension

Upper Dimension

${\displaystyle (xu\equiv xv\land yu\equiv yv\land zu\equiv zv\land u\neq v)\rightarrow (Bxyz\lor Byzx\lor Bzxy).}$

Three points equidistant from two distinct points form a line. Without this axiom, the theory could be modeled by three-dimensional or higher-dimensional space.

Axiom of Euclid

Each of the three variants of this axiom, all equivalent over the remaining Tarski's axioms to Euclid's parallel postulate, has an advantage over the others:

• A dispenses with existential quantifiers;
• B has the fewest variables and atomic sentences;
• C requires but one primitive notion, betweenness. This variant is the usual one given in the literature.

A: ${\displaystyle ((Bxyw\land xy\equiv yw)\land (Bxuv\land xu\equiv uv)\land (Byuz\land yu\equiv zu))\rightarrow yz\equiv vw.}$

Let a line segment join the midpoint of two sides of a given triangle. That line segment will be half as long as the third side. This is equivalent to the interior angles of any triangle summing to two right angles.

B: ${\displaystyle Bxyz\lor Byzx\lor Bzxy\lor \exists a\,(xa\equiv ya\land xa\equiv za).}$

Given any triangle, there exists a circle that includes all of its vertices.

Axiom of Euclid: C

C: ${\displaystyle (Bxuv\land Byuz\land x\neq u)\rightarrow \exists a\,\exists b\,(Bxya\land Bxzb\land Bavb).}$

Given any angle and any point v in its interior, there exists a line segment including v, with an endpoint on each side of the angle.

Five Segment

Five segment

${\displaystyle {(x\neq y\land Bxyz\land Bx'y'z'\land xy\equiv x'y'\land yz\equiv y'z'\land xu\equiv x'u'\land yu\equiv y'u')}\rightarrow zu\equiv z'u'.}$

Begin with two triangles, xuz and x'u'z'. Draw the line segments yu and y'u', connecting a vertex of each triangle to a point on the side opposite to the vertex. The result is two divided triangles, each made up of five segments. If four segments of one triangle are each congruent to a segment in the other triangle, then the fifth segments in both triangles must be congruent.

This is equivalent to the side-angle-side rule for determining that two triangles are congruent; if the angles uxz and u'x'z' are congruent (there exist congruent triangles xuz and x'u'z'), and the two pairs of incident sides are congruent (xu ≡ x'u' and xz ≡ x'z'), then the remaining pair of sides is also congruent (uz ≡ u'z').

Segment Construction

${\displaystyle \exists z\,[Bxyz\land yz\equiv ab].}$

For any point y, it is possible to draw in any direction (determined by x) a line congruent to any segment ab.