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| {{otheruses4|axioms for Euclidean geometry||Tarski–Grothendieck set theory}}
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| '''Tarski's axioms''', due to [[Alfred Tarski]], are an axiom set for the substantial fragment of [[Euclidean geometry]], called "[[elementary theory |elementary]]," that is formulable in [[first-order logic]] with [[identity (mathematics)|identity]], and requiring no [[set theory]] {{harv|Tarski|1959}}. Other modern axiomizations of Euclidean geometry are those by [[Hilbert's axioms|Hilbert]] and [[Birkhoff's axioms|George Birkhoff]]. | |
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| ==Overview==
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| Early in his career Tarski taught geometry and researched set theory. His coworker Steven Givant (1999) explained Tarski's take-off point:
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| :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.
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| Givant's then says "with typical thoroughness" Tarski devised his system:
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| :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 notion]]s 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.
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| Like other modern axiomatizations of Euclidean geometry, Tarski's employs a [[formal system]] consisting of symbol strings, called [[sentence (mathematical logic)|sentence]]s, whose construction respects formal [[syntax (logic)|syntactical rules]], and rules of proof that determine the allowed manipulations of the sentences. Unlike some other modern axiomatizations, such as [[Birkhoff's axioms|Birkhoff's]] and [[Hilbert's axioms|Hilbert's]], Tarski's axiomatization has no [[primitive object]]s 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 [[first-order theory]], it is not even possible to define lines as sets of points. The only primitive relations ([[predicate (mathematical logic)|predicate]]s) are "betweenness" and "congruence" among points.
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| 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'' Euclidian 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 universal-existential form, a special case of the [[prenex normal form]]. This form has all [[Universal quantification|universal quantifiers]] preceding any [[Existential quantification|existential quantifiers]], so that all sentences can be recast in the form <math>\forall u \forall v \ldots\exists a \exists b\dots.</math> This fact allowed Tarski to prove that Euclidean geometry is [[Decidability (logic)|decidable]]: there exists an [[algorithm]] which can determine the truth or falsity of any sentence. Tarski's axiomatization is also [[Completeness|complete]]. This does not contradict [[Gödel's first incompleteness theorem]], because Tarski's theory lacks the expressive power needed to interpret [[Robinson arithmetic]] {{harv|Franzén|2005|pp=25–26}}.
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| == The axioms ==
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| [[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 [[axiom]]s 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.
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| === Fundamental relations ===
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| 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 logic|first-order theory]]. Tarski did so by positing a [[universe (mathematics)|universe]] of [[point (geometry)|point]]s, with lower case letters denoting variables ranging over that universe. [[Equality (mathematics)|Equality]] is provided by the underlying logic (see [[First-order logic#Equality and its axioms]]).<ref>Tarski and Givant, 1999, page 177</ref> Tarski then posited two primitive relations:
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| * ''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'').
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| * ''[[congruence (geometry)|Congruence]]'' (or "equidistance"), a [[polyadic relation|tetradic relation]]. The [[atomic sentence]] ''wx'' ≡ ''yz'' can be interpreted as ''wx'' is [[congruence (geometry)|congruent]] to ''yz'', in other words, that the [[distance|length]] of the line segment ''wx'' is equal to the length of the line segment ''yz''.
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| Betweenness captures the [[affine geometry|affine]] aspect of Euclidean geometry; congruence, its [[metric space|metric]] aspect. The background logic includes [[identity (mathematics)|identity]], a [[binary relation]]. The axioms invoke identity (or its negation) on five occasions.
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| 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 closure]]s; hence any [[free variable]]s should be taken as tacitly [[universal quantifier|universally quantified]].
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| === Congruence axioms ===
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| ; Reflexivity of Congruence:
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| : <math>xy \equiv yx\,.</math>
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| ; Identity of Congruence:
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| : <math>xy \equiv zz \rightarrow x=y.</math>
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| ; [[Transitive relation|Transitivity]] of Congruence:
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| : <math>(xy \equiv zu \and xy \equiv vw) \rightarrow zu \equiv vw.</math>
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| ====Commentary====
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| While the congruence relation <math>xy \equiv zw</math> is, formally, a 4-way relation among points, it may also be thought of, informally, as a binary relation between two line segments <math>xy</math> and <math>zw</math>. The "Reflexivity" and "Transitivity" axioms above, combined, prove both:
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| * that this binary relation is in fact an [[equivalence relation]]
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| ** it is reflexive: <math>xy \equiv xy\,</math>.
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| ** it is symmetric <math>xy \equiv zw \rightarrow zw \equiv xy\,</math>.
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| ** it is transitive <math>(xy \equiv zu \and zu \equiv vw) \rightarrow xy \equiv vw\,</math>.
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| * and that the order in which the points of a line segment are specified is irrelevant.
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| ** <math>xy \equiv zw \rightarrow xy \equiv wz\,</math>.
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| ** <math>xy \equiv zw \rightarrow yx \equiv zw\,</math>.
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| ** <math>xy \equiv zw \rightarrow yx \equiv wz\,</math>.
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| Interestingly, the "Transitivity" axiom asserts that congruence is [[Euclidean relation|Euclidean]], in that it respects the first of [[Euclid's elements|Euclid's]] "[[Euclid's axioms#Axiomatic approach|common notions]]".
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| 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 [[reflexive relation|reflexivity]] for [[binary relation]]s.
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| === Betweenness axioms ===
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| [[File:Tarski's formulation of Pasch's axiom.svg|right|thumb|Pasch's axiom]]
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| ; Identity of Betweenness
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| : <math>Bxyx \rightarrow x=y.</math>
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| The only point on the line segment <math>xx</math> is <math>x</math> itself.
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| <!--
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| ; Transitivity of betweenness
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| : <math>(wxy \and xyz) \rArr wxz</math>
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| If ''x'' is between ''w'' and ''y'', and if ''y'' is between ''x'' and ''z'', then ''x'' must be between ''w'' and ''z''.
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| ; Connectivity of betweenness
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| : <math>(xyz \and xyw \and x \ne y) \rArr (xzw \or xwz).</math>
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| If ''y'' is between ''x'' and ''z'' and between ''x'' and ''w'', ''w'' and ''z'' must both be on the same side of ''x''. Connectivity and Transitivity order the points.-->
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| ; [[Axiom of Pasch]]
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| : <math>(Bxuz \and Byvz) \rightarrow \exists a\, (Buay \and Bvax).</math>
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| The two diagonals of the [[quadrilateral]] <math>xuvy</math> must intersect at some point.
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| [[File:Tarski's continuity axiom.svg|right|thumb|Continuity: φ and ψ divide the ray into two halves and the axiom asserts the existence of a point b dividing those two halves]]
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| ; [[Axiom schema]] of Continuity
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| Let φ(''x'') and ψ(''y'') be [[first order logic|first-order formulae]] containing no [[free variable|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:
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| : <math>\exists a \,\forall x\, \forall y\,[(\phi(x) \and \psi(y)) \rightarrow Baxy] \rightarrow \exists b\, \forall x\, \forall y\,[(\phi(x) \and \psi(y)) \rightarrow Bxby].</math>
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| 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.
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| ; Lower [[Dimension]]
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| : <math>\exists a \, \exists b\, \exists c\, [\neg Babc \and \neg Bbca \and \neg Bcab].</math>
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| There exist three noncollinear points. Without this axiom, the theory could be [[model theory|modeled]] by the one-dimensional [[real line]], a single point, or even the empty set.
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| === Congruence and betweenness ===
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| [[File:Points in a plane equidistant to two given points lie on a line.svg|right|thumb|Upper dimension]]
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| ; Upper [[Dimension]]
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| : <math>(xu \equiv xv \and yu \equiv yv \and zu \equiv zv \and u \ne v) \rightarrow (Bxyz \or Byzx \or Bzxy).</math>
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| Three points equidistant from two distinct points form a line. Without this axiom, the theory could be modeled by [[three-dimensional space|three-dimensional]] or higher-dimensional space.
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| ; Axiom of Euclid
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| 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:
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| * '''A''' dispenses with [[existential quantifier]]s;
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| * '''B''' has the fewest variables and [[atomic sentence]]s;
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| * '''C''' requires but one primitive notion, betweenness. This variant is the usual one given in the literature.
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| : '''A''': <math>((Bxyw \and xy \equiv yw ) \and (Bxuv \and xu \equiv uv) \and (Byuz \and yu \equiv zu)) \rightarrow yz \equiv vw.</math>
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| 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 angle]]s of any triangle summing to two [[right angles]].
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| : '''B''': <math>Bxyz \or Byzx \or Bzxy \or \exists a\, (xa \equiv ya \and xa \equiv za).</math>
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| Given any [[triangle]], there exists a [[circle]] that includes all of its vertices.
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| [[File:Tarski's axiom of Euclid C.svg|thumb|right|Axiom of Euclid: C]]
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| : '''C''': <math>(Bxuv \and Byuz \and x \ne u) \rightarrow \exists a\, \exists b\,(Bxya \and Bxzb \and Bavb).</math>
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| 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.
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| ; Five Segment
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| [[File:Five segment.svg|thumb|right|Five segment]]
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| :<math>{(x \ne y \and Bxyz \and Bx'y'z' \and xy \equiv x'y' \and yz \equiv y'z' \and xu \equiv x'u' \and yu \equiv y'u')} \rightarrow zu \equiv z'u'.</math>
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| Begin with two [[triangle]]s, ''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 [[congruence (geometry)|congruent]] to a segment in the other triangle, then the fifth segments in both triangles must be congruent.
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| ; Segment Construction
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| : <math> \exists z\, [Bxyz \and yz \equiv ab].</math>
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| For any point ''y'', it is possible to draw in some direction (determined by ''x'') a line congruent to any segment ''ab''.
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| ==Discussion==
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| Starting from two primitive [[Relation (mathematics)|relations]] whose fields are a [[density|dense]] [[universe (mathematics)|universe]] of [[point (geometry)|point]]s, Tarski built a geometry of [[line segment]]s. According to Tarski and Givant (1999: 192-93), none of the above [[axiom]]s 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.
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| A number of other properties of Betweenness are derivable as theorems including:
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| *[[Reflexive relation|Reflexivity]]: ''Bxxy'' ;
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| *[[Symmetry]]: ''Bxyz'' → ''Bzyx'' ;
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| *[[Transitive relation|Transitivity]]: (''Bxyw'' ∧ ''Byzw'') → ''Bxyz'' ;
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| *[[Connection (mathematics)|Connectivity]]: (''Bxyw'' ∧ ''Bxzw'') → (''Bxyz'' ∨ ''Bxzy'').
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| The last two properties [[total order|totally order]] the points making up a line segment.
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| Upper and Lower Dimension together require that any model of these axioms have a specific finite [[dimension]]ality. Suitable changes in these axioms yield axiom sets for [[Euclidean geometry]] for [[dimension]]s 0, 1, and greater than 2 (Tarski and Givant 1999: Axioms 8<sup>(1)</sup>, 8<sup>(n)</sup>, 9<sup>(0)</sup>, 9<sup>(1)</sup>, 9<sup>(n)</sup> ). 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.
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| When dimension > 1, Betweenness can be defined in terms of [[congruence relation|congruence]] (Tarski and Givant, 1999). First define the relation "≤" (where <math>ab \leq cd</math> is interpreted "the length of line segment <math>ab</math> is less than or equal to the length of line segment <math>cd</math>"):
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| :<math>xy \le zu \leftrightarrow \forall v ( zv \equiv uv \rightarrow \exists w ( xw \equiv yw \and yw \equiv uv ) ).</math>
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| 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''.
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| Betweenness can than be defined by using the intuition that the shortest distance between any two points is a straight line:
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| :<math>Bxyz \leftrightarrow \forall u ( ( ux \le xy \and uz \le zy ) \rightarrow u = y ).</math>
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| The Axiom Schema of Continuity assures that the ordering of points on a line is [[Dedekind complete|complete]] (with respect to first-order definable properties). The Axioms of [[Pasch's axiom|Pasch]] and Euclid are well known. Remarkably, Euclidean geometry requires just the following further axioms:
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| * ''Segment Construction''. This axiom makes [[measurement]] and the [[Cartesian coordinate system]] possible—simply assign the value of 1 to some arbitrary non-empty line segment;{{clarify|date=December 2011}}<!-- what would this axiom look like? -->
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| <!-- Already included above * ''Five Segments''. This bears on the [[congruence (geometry)|congruence]] of [[triangle]]s.{{Contradict|date=May 2012}}-->
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| 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:
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| *[[consistency|Consistent]]: There is no wff such that it and its negation are both theorems;
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| *[[complete theory|Complete]]: Every sentence or its negation is a theorem provable from the axioms;
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| *[[Decidability (logic)|Decidable]]: There exists an [[algorithm]] that assigns a [[truth value]] to every sentence. This follows from Tarski's:
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| ** [[Decision procedure]] for the [[real closed field]], which he found by [[quantifier elimination]];
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| **Axioms admitting of a (multi-dimensional) faithful [[interpretability|interpretation]] as a [[real closed field]].
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| Gupta (1965) proved the above axioms independent, ''Pasch'' and ''Reflexivity of Congruence'' excepted.
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| 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.
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| ==Comparison with Hilbert==
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| [[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. 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 logic|first-order theory]]. Hilbert's axioms do not constitute a first-order theory because his continuity axioms require [[second-order logic]].
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| The first four groups of axioms of [[Hilbert's axioms]] for plane geometry can be proved using Tarski's axioms.
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| ==Notes==
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| <references/>
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| ==References==
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| *{{citation
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| | last1=Franzén
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| | first1=Torkel
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| | author1-link=Torkel Franzén
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| | year = 2005
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| | title =Gödel's Theorem: An Incomplete Guide to Its Use and Abuse
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| | publisher= A K Peters
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| | isbn=1-56881-238-8
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| }}
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| *Givant, Steven (1999) "Unifying threads in Alfred Tarski's Work", [[Mathematical Intelligencer]] 21:47–58.
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| *Gupta, H. N. (1965) ''Contributions to the Axiomatic Foundations of Geometry''. Ph.D. thesis, University of California-Berkeley.
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| *{{Citation
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| | last1=Tarski
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| | first1=Alfred
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| | author1-link=Alfred Tarski
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| | editor1-last=Leon Henkin, Patrick Suppes and Alfred Tarski
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| | title=The axiomatic method. With special reference to geometry and physics. Proceedings of an International Symposium held at the Univ. of Calif., Berkeley, Dec. 26, 1957-Jan. 4, 1958
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| | publisher=North-Holland
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| | location=Amsterdam
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| | series=Studies in Logic and the Foundations of Mathematics
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| | id={{MathSciNet | id = 0106185}}
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| | year=1959
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| | chapter=What is elementary geometry?
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| | pages=16–29}}. Available as a 2007 [http://books.google.com/books?id=eVVKtnKzfnUC&pg=PA16 reprint], Brouwer Press, ISBN 1-4437-2812-8
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| *{{Citation
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| | doi=10.2307/421089
| |
| | last1=Tarski
| |
| | first1=Alfred
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| | author1-link=Alfred Tarski
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| | last2=Givant
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| | first2=Steven
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| | title=Tarski's system of geometry
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| | url=http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.27.9012
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| | id={{MathSciNet | id = 1791303}}
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| | year=1999
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| | journal=The Bulletin of Symbolic Logic
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| | issn=1079-8986
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| | volume=5
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| | issue=2
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| | pages=175–214
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| | jstor=421089}}
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| *Schwabhäuser, W., Szmielew, W., and [[Alfred Tarski]], 1983. ''Metamathematische Methoden in der Geometrie''. Springer-Verlag.
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| *Szczerba, L. W., 1986, "Tarski and Geometry," ''Journal of Symbolic Logic'' 51: 907-12.
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| [[Category:Elementary geometry]]
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| [[Category:Axiomatics of Euclidean geometry]]
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| [[Category:Mathematical axioms]]
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