Tarski's axioms are an axiom system for Euclidean geometry, specifically for that portion of Euclidean geometry that is formulable in firstorder logic with identity (i.e. is formulable as an elementary theory). As such, it does not require an underlying set theory. The only primitive objects of the system are "points" and the only primitive predicates are "betweenness" (expressing the fact that a point lies on a line segment between two other points) and "congruence" (expressing the fact that the distance between two points equals the distance between two other points). The system contains infinitely many axioms.
The axiom system is due to Alfred Tarski who first presented it in 1926.^{[1]} Other modern axiomizations of Euclidean geometry are Hilbert's axioms (1899) and Birkhoff's axioms (1932).
Using his axiom system, Tarski was able to show that the firstorder theory of Euclidean geometry is consistent, complete and decidable: every sentence in its language is either provable or disprovable from the axioms, and we have an algorithm which decides for any given sentence whether it is provable or not.
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Overview
Early in his career Tarski taught geometry and researched set theory. His coworker Steven Givant (1999) explained Tarski's takeoff point:
 From Enriques, Tarski learned of the work of Mario Pieri, an Italian geometer who was strongly influenced by Peano. Tarski preferred Pieri's system [of his Point and Sphere memoir], where the logical structure and the complexity of the axioms were more transparent.
Givant then says that "with typical thoroughness" Tarski devised his system:
 What was different about Tarski's approach to geometry? First of all, the axiom system was much simpler than any of the axiom systems that existed up to that time. In fact the length of all of Tarski's axioms together is not much more than just one of Pieri's 24 axioms. It was the first system of Euclidean geometry that was simple enough for all axioms to be expressed in terms of the primitive notions only, without the help of defined notions. Of even greater importance, for the first time a clear distinction was made between full geometry and its elementary — that is, its first order — part.
Like other modern axiomatizations of Euclidean geometry, Tarski's employs a formal system consisting of symbol strings, called sentences, whose construction respects formal syntactical rules, and rules of proof that determine the allowed manipulations of the sentences. Unlike some other modern axiomatizations, such as Birkhoff's and Hilbert's, Tarski's axiomatization has no primitive objects other than points, so a variable or constant cannot refer to a line or an angle. Because points are the only primitive objects, and because Tarski's system is a firstorder theory, it is not even possible to define lines as sets of points. The only primitive relations (predicates) are "betweenness" and "congruence" among points.
Tarski's axiomatization is shorter than its rivals, in a sense Tarski and Givant (1999) make explicit. It is more concise than Pieri's because Pieri had only two primitive notions while Tarski introduced three: point, betweenness, and congruence. Such economy of primitive and defined notions means that Tarski's system is not very convenient for doing Euclidean geometry. Rather, Tarski designed his system to facilitate its analysis via the tools of mathematical logic, i.e., to facilitate deriving its metamathematical properties. Tarski's system has the unusual property that all sentences can be written in universalexistential form, a special case of the prenex normal form. This form has all universal quantifiers preceding any existential quantifiers, so that all sentences can be recast in the form This fact allowed Tarski to prove that Euclidean geometry is decidable: there exists an algorithm which can determine the truth or falsity of any sentence. Tarski's axiomatization is also complete. This does not contradict Gödel's first incompleteness theorem, because Tarski's theory lacks the expressive power needed to interpret Robinson arithmetic (Franzén 2005, pp. 25–26).
The axioms
Alfred Tarski worked on the axiomatization and metamathematics of Euclidean geometry intermittently from 1926 until his death in 1983, 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 firstorder 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 Firstorder logic#Equality and its axioms).^{[2]} Tarski then posited two primitive relations:
 Betweenness, a triadic relation. The atomic sentence Bxyz denotes that the point y is "between" the points 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 Cwxyz or commonly 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 (such as the parallelism of lines) of Euclidean geometry; congruence, its metric aspect (such as angles and distances). The background logic includes identity, a binary relation denoted by =.
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.
Congruence axioms
 Reflexivity of Congruence
 Identity of Congruence
 Transitivity of Congruence
Commentary
While the congruence relation is, formally, a 4way relation among points, it may also be thought of, informally, as a binary relation between two line segments and . The "Reflexivity" and "Transitivity" axioms above, combined, prove both:
 that this binary relation is in fact an equivalence relation
 it is reflexive: .
 it is symmetric .
 it is transitive .
 and that the order in which the points of a line segment are specified is irrelevant.
 .
 .
 .
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.
Betweenness axioms
 Identity of Betweenness
The only point on the line segment is itself.
 Axiom schema of Continuity
Let φ(x) and ψ(y) be firstorder 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.
Note that the formulae φ(x) and ψ(y) may contain parameters, i.e. free variables different from a, b, x, y. And indeed, each instance of the axiom scheme that does not contain parameters can be proven from the other axioms.^{[3]}
 Lower Dimension
There exist three noncollinear points. Without this axiom, the theory could be modeled by the onedimensional real line, a single point, or even the empty set.
Congruence and betweenness
 Upper Dimension
Three points equidistant from two distinct points form a line. Without this axiom, the theory could be modeled by threedimensional or higherdimensional space.
 Axiom of Euclid
Three variants of this axiom can be given, labeled A, B and C below. They are equivalent to each other given the remaining Tarski's axioms, and indeed equivalent to Euclid's parallel postulate.
 A:
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:
Given any triangle, there exists a circle that includes all of its vertices.
 C:
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.
Each variant 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.
 Five Segment
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 sideangleside 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
For any point y, it is possible to draw in any direction (determined by x) a line congruent to any segment ab.
Discussion
According to Tarski and Givant (1999: 19293), none of the above axioms are 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^{[4]} 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.
The Upper and Lower Dimension axioms together require that any model of these axioms have dimension 2, i.e. that we are axiomatizing the Euclidean plane. 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 the number of dimensions is greater than 1, Betweenness can be defined in terms of congruence (Tarski and Givant, 1999). First define the relation "≤" (where is interpreted "the length of line segment is less than or equal to the length of line segment "):
In the case of two dimensions, the intuition is as follows: For any line segment xy, consider the possible range of lengths of xv, where v is any point on the perpendicular bisector of xy. It is apparent that while there is no upper bound to the length of xv, there is a lower bound, which occurs when v is the midpoint of xy. So if xy is shorter than or equal to zu, then the range of possible lengths of xv will be a superset of the range of possible lengths of zw, where w is any point on the perpendicular bisector of zu.
Betweenness can then be defined by using the intuition that the shortest distance between any two points is a straight line:
The Axiom Schema of Continuity assures that the ordering of points on a line is complete (with respect to firstorder definable properties). As was pointed out by Tarski, this firstorder axiom schema may be replaced by a more powerful secondorder Axiom of Continuity if one allows for variables to refer to arbitrary sets of points. The resulting secondorder system is equivalent to Hilbert's set of axioms. (Tarski and Givant 1999)
The Axioms of Pasch and Euclid are well known. The Segment Construction axiom makes measurement and the Cartesian coordinate system possible—simply assign the length 1 to some arbitrary nonempty line segment. Indeed, it is shown in (Schwabhäuser 1983) that by specifying two distinguished points on a line, called 0 and 1, we can define an addition, multiplication and ordering, turning the set of points on that line into a realclosed ordered field. We can then introduce coordinates from this field, showing that every model of Tarski's axioms is isomorphic to the twodimensional plane over some realclosed ordered field.
The standard geometric notions of parallelism and intersection of lines (where lines are represented by two distinct points on them), right angles, congruence of angles, similarity of triangles, tangency of lines and circles (represented by a center point and a radius) can all be defined in Tarski's system.
Let wff stand for a wellformed formula (or syntactically correct firstorder formula) in Tarski's system. Tarski and Givant (1999: 175) proved that Tarski's system is:
 Consistent: There is no wff such that it and its negation can both be proven from the axioms;
 Complete: Every wff or its negation is a theorem provable from the axioms;
 Decidable: There exists an algorithm that decides for every wff whether is it is provable or disprovable from the axioms. This follows from Tarski's:
 Decision procedure for the real closed field, which he found by quantifier elimination (the Tarski–Seidenberg theorem);
 Axioms admitting the abovementioned representation as a twodimensional plane over a real closed field.
This has the consequence that every statement of (secondorder, general) Euclidean geometry which can be formulated as a firstorder sentence in Tarski's system is true if and only if it is provable in Tarski's system, and this provability can be automatically checked with Tarski's algorithm. This, for instance, applies to all theorems in Euclid's Elements, Book I. An example of a theorem of Euclidean geometry which cannot be so formulated is the Archimedean property: to any two positivelength line segments S_{1} and S_{2} there exists a natural number n such that nS_{1} is longer than S_{2}. (This is a consequence of the fact that there are realclosed fields that contain infinitesimals.^{[5]}) Other notions that cannot be expressed in Tarski's system are the constructability with straightedge and compass and statements that talk about "all polygones" etc.^{[6]}
Gupta (1965) proved the Tarski's axioms independent, excepting Pasch and Reflexivity of Congruence.
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's system
Hilbert's axioms for plane geometry number 16, 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.
Hilbert uses two axioms of Continuity, and they require secondorder logic. By contrast, Tarski's Axiom schema of Continuity consists of infinitely many firstorder axioms. Such a schema is indispensable; Euclidean geometry in Tarski's (or equivalent) language cannot be finitely axiomatized as a firstorder theory.
Hilbert's system is therefore considerably stronger: every model is isomorphic to the real plane (using the standard notions of points and lines). By contrast, Tarski's system has many nonisomorphic models: for every realclosed field F, the plane F^{2} provides one such model (where betweenness and congruence are defined in the obvious way).^{[7]}
The first four groups of axioms of Hilbert's axioms for plane geometry are biinterpretable with Tarski's axioms minus continuity.
See also
Notes
 ^ Tarski 1959, Tarski and Givant 1999
 ^ Tarski & Givant 1999, p. 177.
 ^ Schwabhäuser 1983, p. 287288
 ^ Tarski and Givant 1999, p. 189
 ^ Greenberg 2010
 ^ McNaughton, Robert (1953). "Review: A decision method for elementary algebra and geometry by A. Tarski" (PDF). Bull. Amer. Math. Soc. 59 (1): 91–93. doi:10.1090/s000299041953096641.
 ^ Schwabhäuser 1983, section I.16
References
 Franzén, Torkel (2005), Gödel's Theorem: An Incomplete Guide to Its Use and Abuse, A K Peters, ISBN 1568812388
 Givant, Steven (1 December 1999). "Unifying threads in Alfred Tarski's Work". The Mathematical Intelligencer. 21 (1): 47–58. doi:10.1007/BF03024832. ISSN 18667414. S2CID 119716413.
 Greenberg, Marvin Jay (2010). "Old and New Results in the Foundations of Elementary Plane Euclidean and NonEuclidean Geometries" (PDF). The American Mathematical Monthly. 117 (3): 198. doi:10.4169/000298910x480063.
 Gupta, H. N. (1965). Contributions to the Axiomatic Foundations of Geometry (Ph.D. thesis). University of CaliforniaBerkeley.
 Tarski, Alfred (1959), "What is elementary geometry?", in Leon Henkin, Patrick Suppes and Alfred Tarski (ed.), The axiomatic method. With special reference to geometry and physics. Proceedings of an International Symposium held at the Univ. of Calif., Berkeley, Dec. 26, 1957Jan. 4, 1958, Studies in Logic and the Foundations of Mathematics, Amsterdam: NorthHolland, pp. 16–29, MR 0106185.
 Available as a 2007 reprint, Brouwer Press, ISBN 1443728128
 Tarski, Alfred; Givant, Steven (1999), "Tarski's system of geometry", The Bulletin of Symbolic Logic, 5 (2): 175–214, CiteSeerX 10.1.1.27.9012, doi:10.2307/421089, ISSN 10798986, JSTOR 421089, MR 1791303, S2CID 18551419
 Schwabhäuser, W.; Szmielew, W.; Tarski, Alfred (1983). Metamathematische Methoden in der Geometrie. SpringerVerlag.
 Szczerba, L. W. (1986). "Tarski and Geometry" (PDF). Journal of Symbolic Logic. 51 (4): 907–12. doi:10.2307/2273904. JSTOR 2273904. S2CID 35275962.