Tarski's axioms: Difference between revisions

{{otheruses4|axioms for Euclidean geometry|Tarski's axioms for the [[real numbers]]|Tarski's axiomatization of the reals|Tarski's axioms for [[set theory]]|Tarski–Grothendieck set theory}}

Givant's then says that "with typical thoroughness" Tarski devised his system:

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>{{sfn|Tarski and |Givant, |1999, |page =177</ref> }} 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 (geometry)|Congruence]]'' (or "equidistance"), a [[polyadic relation|tetradic relation]]. The [[atomic sentence]] ''Cwxyz'' or commonly ''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''.

Betweenness captures the [[affine geometry|affine]] aspect (such as the parallelism of lines) of Euclidean geometry; congruence, its [[metric space|metric]] aspect (such as angles and distances). The background logic includes [[identity (mathematics)|identity]], a [[binary relation]]. Thedenoted axioms invoke identityby (or=. its negation) on five occasions.

: <math>(xy \equiv zu \andland xy \equiv vw) \rightarrow zu \equiv vw.</math>

** it is reflexive: <math>xy \equiv xy\,</math>.

** it is symmetric <math>xy \equiv zw \rightarrow zw \equiv xy\,</math>.

** it is transitive <math>(xy \equiv zu \andland zu \equiv vw) \rightarrow xy \equiv vw\,</math>.

** <math>xy \equiv zw \rightarrow xy \equiv wz\,</math>.

** <math>xy \equiv zw \rightarrow yx \equiv zw\,</math>.

** <math>xy \equiv zw \rightarrow yx \equiv wz\,</math>.

Interestingly, theThe "Transitivitytransitivity" 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]]".

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.

: <math>(wxy \andland xyz) \rArr wxz</math>

: <math>(xyz \andland xyw \andland x \ne y) \rArr (xzw \orlor xwz).</math>

: <math>(Bxuz \andland Byvz) \rightarrow \exists a\, (Buay \andland Bvax).</math>

: <math>\exists a \,\forall x\, \forall y\,[(\phi(x) \andland \psi(y)) \rightarrow Baxy] \rightarrow \exists b\, \forall x\, \forall y\,[(\phi(x) \andland \psi(y)) \rightarrow Bxby].</math>

: <math>\exists a \, \exists b\, \exists c\, [\neg Babc \andland \neg Bbca \andland \neg Bcab].</math>

[[File:Points in a plane equidistant to two given points lie on a line.svg|right|thumb|Upper dimension axiom]]

: <math>(xu \equiv xv) \andland (yu \equiv yv) \andland (zu \equiv zv )\andland (u \ne v) \rightarrow (Bxyz \orlor Byzx \orlor Bzxy).</math>

Each of the threeThree variants of this axiom can be given, alllabeled A, B and C below. They are equivalent overto each other given the remaining Tarski's axioms, and indeed equivalent to Euclid's [[parallel postulate]], has an advantage over the others:.

: '''B''': <math>Bxyz \orlor Byzx \orlor Bzxy \orlor \exists a\, (xa \equiv ya \andland xa \equiv za).</math>

: '''C''': <math>(Bxuv \andland Byuz \andland x \ne u) \rightarrow \exists a\, \exists b\,(Bxya \andland Bxzb \andland Bavb).</math>

:<math>{(x \ne y \andland Bxyz \andland Bx'y'z' \andland xy \equiv x'y' \andland yz \equiv y'z' \andland xu \equiv x'u' \andland yu \equiv y'u')} \rightarrow zu \equiv z'u'.</math>

: <math> \exists z\, [Bxyz \andland yz \equiv ab].</math>

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

Starting from two primitive [[Relation (mathematics)|relations]] whose fields are a [[Dense-in-itself|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 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<ref>Tarski and Givant 1999, p. 189</ref> including:

The Upper and Lower Dimension axioms together require that any model of these axioms have adimension specific2, finitei.e. [[dimension]]alitythat we are axiomatizing the Euclidean plane. 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⁽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 dimensionthe >number of dimensions is greater than 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>"):

:<math>xy \le zu \leftrightarrow \forall v ( zv \equiv uv \rightarrow \exists w ( xw \equiv yw \andland yw \equiv uv ) ).</math>

:<math>Bxyz \leftrightarrow \forall u ( ( ux \le xy \andland uz \le zy ) \rightarrow u = y ).</math>

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). TheAs Axiomswas ofpointed out by Tarski, this first-order axiom schema may be replaced by a more powerful [[Pasch'sSecond-order axiomlogic|Paschsecond-order]] andAxiom Euclidof areContinuity wellif knownone allows for variables to refer to arbitrary sets of points. Remarkably,The Euclideanresulting geometrysecond-order requiressystem justis theequivalent followingto furtherHilbert's set of axioms:. (Tarski and Givant 1999)

**Axioms admitting ofthe above-mentioned representation as a (multitwo-dimensional) faithfulplane [[interpretability|interpretation]] asover a [[real closed field]].

==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. The

Hilbert [[Axiomuses schema]]two axioms of Continuity, playsand athey rolerequire similar[[second-order tologic]]. HilbertBy contrast, Tarski's two[[Axiom axiomsschema]] of Continuity consists of infinitely many first-order axioms. ThisSuch a 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]].

*{{citation |last1 = Franzén

|author1-link first1= Torkel Franzén

*{{cite journal |last1=Givant, |first1=Steven (1999) "|title=Unifying threads in Alfred Tarski's Work", |journal=[[The Mathematical Intelligencer]] |date=1 December 1999 |volume=21 |issue=1 |pages=47–58 |doi=10.1007/BF03024832 |s2cid=119716413 |url=https:47&ndash;58//link.springer.com/article/10.1007/BF03024832 |access-date= |language=en |issn=1866-7414}}

| pages=16–29}}.
**Available as a 2007 [https://books.google.com/books?id=eVVKtnKzfnUC&pg=PA16 reprint], Brouwer Press, {{isbn|1-4437-2812-8}}