Tarski's axioms

Tarski's axioms, due to Alfred Tarski, are an axiom set for the substantial fragment of Euclidean geometry, called "elementary," that is formulable in first-order logic with identity, and requiring no set theory. Other modern axiomizations of Euclidean geometry are those by Hilbert and George Birkhoff.

The axioms

Alfred Tarski worked on the axiomatization and metamathematics of Euclidean geometry intermittently from 1926 until his 1983 death, with Tarski (1959) heralding his mature interest in the subject. The work of Tarski and his students on Euclidean geometry culminated in the monograph Schwabhäuser, Szmielew, and Tarski (1983), which set out the 10 axioms and one axiom schema shown below, the associated metamathematics, and a fair bit of the subject. Gupta (1965) made important contributions, and Tarski and Givant (1999) discuss the history.

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. He then posited two primitive relations:
* "Betweeness", a triadic relation. The atomic sentence "Bxyz" denotes that "y" is "between" "x" and "z", i.e., that "x", "y", and "z" are collinear with "y" "between" them;
* "Congruence" (or "equidistance"), a tetradic relation. Let "xy" denote the line segment whose endpoints are "x" and "y". The atomic sentence "wx" ≡ "yz" has two intuitive meanings:
** "wx" is congruent to "yz";
** The distance from "w" to "x" equals the distance from "y" to "z".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 univeral closures; hence any free variables should be taken as tacitly universally quantified.

Congruence axioms

; Reflexivity of Congruence:: $xy\; equiv\; yx,.$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:: $xy\; equiv\; zz\; ightarrow\; x=y.$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:: $(xy\; equiv\; zu\; and\; xy\; equiv\; vw)\; ightarrow\; zu\; equiv\; vw.$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: $Bxyx\; ightarrow\; x=y.$It is not possible for a point to be "between" a point; points are indivisible.

; Axiom of Pasch: $(Bxuz\; and\; Byvz)\; ightarrow\; exists\; a,\; (Buay\; and\; Bvax).$ 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 ContinuityLet 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:: $exists\; a\; ,forall\; x,\; forall\; y,\; [(phi(x)\; and\; psi(y))\; ightarrow\; Baxy]\; ightarrow\; exists\; b,\; forall\; x,\; forall\; y,\; [(phi(x)\; and\; psi(y))\; ightarrow\; Bxby]\; .$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".

; Lower Dimension: $exists\; a\; ,\; exists\; b,\; exists\; c,\; [\; eg\; Babc\; and\; eg\; Bbca\; and\; eg\; Bcab]\; .$In short, there exist three noncollinear points, and any model of these axioms must have dimension > 1.

Congruence and betweenness

; Upper Dimension: $(xu\; equiv\; xv\; and\; yu\; equiv\; yv\; and\; zu\; equiv\; zv\; and\; u\; e\; v)\; ightarrow\; (Bxyz\; or\; Byzx\; or\; Bzxy).$Three points equidistant from two distinct points form a line. Hence any model of these axioms must have dimension < 3.

; Axiom of EuclidEach of the three variants of this axiom, all equivalent 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: $((Bxyw\; and\; xy\; equiv\; yw\; )\; and\; (Bxuv\; and\; xu\; equiv\; uv)\; and\; (Byuz\; and\; yu\; equiv\; zu))\; ightarrow\; yz\; equiv\; vw.$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: $Bxyz\; or\; Byzx\; or\; Bzxy\; or\; exists\; a,\; (xa\; equiv\; ya\; and\; xa\; equiv\; za).$Given any triangle, there exists a circle that includes all of its vertices.

: C: $(Bxuv\; and\; Byuz\; and\; x\; e\; u)\; ightarrow\; exists\; a,\; exists\; b,(Bxya\; and\; Bxzb\; and\; Bavb).$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:$\{(x\; e\; y\; and\; Bxyz\; and\; Bx\text{'}y\text{'}z\text{'}\; and\; xy\; equiv\; x\text{'}y\text{'}\; and\; yz\; equiv\; y\text{'}z\text{'}\; and\; xu\; equiv\; x\text{'}u\text{'}\; and\; yu\; equiv\; y\text{'}u\text{'})\}\; ightarrow\; zu\; equiv\; z\text{'}u\text{'}.$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.

; Segment Construction: $exists\; a,\; [Bwxa\; and\; xa\; equiv\; yz]\; .$Given any two line segments, the second can be "extended" by a line segment congruent to the first.

Discussion

Starting from two primitive relations whose fields are a dense universe of points, Tarski built a geometry of line segments. According to Tarski and Givant (1999: 192-93), none of the above axioms is fundamentally new. The first four axioms establish some elementary properties of the two primitive relations. For instance, Reflexivity and Transitivity of Congruence establish that congruence is an equivalence relation over line segments. The Identity of Congruence and of Betweenness govern the trivial case when those relations are applied to nondistinct points. The theorem "xy"≡"zz" ↔ "x"="y" ↔ "Bxyx" extends these Identity axioms.

A number of other properties of Betweenness are derivable as theorems including:
*Reflexivity: "Bxxy" ;
*Symmetry: "Bxyz" → "Bzyx" ;
*Transitivity: ("Bxyw" ∧ "Byzw") → "Bxyz" ;
*Connectivity: ("Bxyw" ∧ "Bxzw") → ("Bxyz" ∨ "Bxzy").The last two properties totally order the points making up a line segment.

Upper and Lower Dimension together require that any model of these axioms have a specific finite dimensionality. Suitable changes in these axioms yield axiom sets for Euclidean geometry for dimensions 0, 1, and greater than 2 (Tarski and Givant 1999: Axioms 8⁽¹⁾, 8⁽ⁿ⁾, 9⁽⁰⁾, 9⁽¹⁾, 9⁽ⁿ⁾ ). Note that solid geometry requires no new axioms, unlike the case with Hilbert's axioms. Moreover, Lower Dimension for "n" dimensions is simply the negation of Upper Dimension for "n" - 1 dimensions.

When dimension > 1, Betweenness can be defined in terms of congruence (Tarski and Givant, 1999). First define the relation "≤" in terms of Congruence::$xy\; le\; zu\; leftrightarrow\; forall\; v\; (\; zv\; equiv\; uv\; ightarrow\; exists\; w\; (\; xw\; equiv\; yw\; and\; yw\; equiv\; uv\; )\; ).$In the case of two dimensions, the intuition is as follows. For all points "v" on the perpendicular bisector of "zu", there is a point "w" on the perpendicular bisector of "xy" such that "yw" is congruent to "vu".

Betweenness can than be defined as:$Bxyz\; leftrightarrow\; forall\; u\; (\; ux\; le\; xy\; and\; uz\; le\; zy\; ightarrow\; u\; =\; y\; ).$The Axiom Schema of Continuity assures that the ordering of points on a line is complete (with respect to first-order definable properties). The Axioms of Pasch and Euclid are well known. Remarkably, Euclidean geometry requires but two more axioms:
* "Segment Construction". This axiom makes measurement and the Cartesian coordinate system possible&mdash;simply assign the value of 1 to some arbitrary line segment;
* "Five Segments". This bears on the congruence of triangles.

Let "wff" stand for a well-formed formula (or syntactically correct formula) of elementary geometry. Tarski and Givant (1999: 175) proved that elementary geometry is:
*Consistent: There is no wff such that it and its negation are both theorems;
*Complete: Every wff or its negation is a theorem provable from the axioms;
*Decidable: There exists an algorithm that assigns a truth value to every wff. This follows from Tarski's:
** Decision procedure for the real closed field, which he found by quantifier elimination;
**Axioms admitting of a (multi-dimensional) faithful interpretation as a real closed field.Gupta (1965) proved the above axioms independent, "Pasch" and "Reflexivity of Congruence" excepted.

Negating the Axiom of Euclid yields hyperbolic geometry, while eliminating it outright yields absolute geometry. Full (as opposed to elementary) Euclidean geometry requires giving up a first order axiomatization: replace φ("x") and ψ("y") in the axiom schema of Continuity with "x" ∈ "A" and "y" ∈ "B", where "A" and "B" are universally quantified variables ranging over sets of points.

Comparison with Hilbert

Hilbert's axioms for plane geometry number 14, and include Transitivity of Congruence and a variant of the Axiom of Pasch. The only notion from intuitive geometry invoked in the remarks to Tarski's axioms is triangle. (Versions B and C of the Axiom of Euclid refer to '"circle" and "angle," respectively.) Hilbert's axioms also require "ray," "angle," and the notion of a triangle "including" an angle. In addition to betweenness and congruence, Hilbert's axioms require a primitive binary relation "on," linking a point and a line. The Axiom schema of Continuity plays a role similar to Hilbert's two axioms of Continuity. This schema is indispensable; Euclidean geometry in Tarski's (or equivalent) language cannot be finitely axiomatized as a first-order theory. Hilbert's axioms do not constitute a first-order theory because his continuity axioms require second-order logic.

References

*Gupta, H. N., 1965. "Contributions to the Axiomatic Foundations of Geometry". Ph.D. thesis, University of California-Berkeley.
*Alfred Tarski, 1959, '"What is Elementary Geometry?" in Leon Henkin, Patrick Suppes, and Tarski, A., eds., "The Axiomatic Method, with Special Reference to Geometry and Physics". North Holland.
*------, and Givant, Steven, 1999, "Tarski's system of geometry," "Bulletin of Symbolic Logic" 5: 175-214. [http://citeseer.ist.psu.edu/tarski99tarskis.html]
*Schwabhäuser, W., Szmielew, W., and Alfred Tarski, 1983. "Metamathematische Methoden in der Geometrie". Springer-Verlag.
*Szczerba, L. W., 1986, "Tarski and Geometry," "Journal of Symbolic Logic" 51: 907-12.