Lumiste - Tarski's system of Geometry and Betweenness Geometry with the Group of Movements.pdf

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Proc. Estonian Acad. Sci. Phys. Math., 2007, 56, 3, 252–263
Tarski’s system of geometry and betweenness
geometry with the group of movements
Ülo Lumiste
Institute of Pure Mathematics, Faculty of Mathematics and Computer Science, University of
Tartu, J. Liivi 2, 50409 Tartu, Estonia; lumiste@math.ut.ee
Received 21 May 2007, in revised form 12 June 2007
Abstract. Recently, in a paper by Tarski and Givant ( Bull. Symbolic Logic , 1999, 5, 175–214),
Tarski’s system of geometry was revived. The system originated in Tarski’s lectures of 1926–
27, but was published in the 1950s–60s and in 1983. On the other hand, the author’s papers of
2005–07 revived the betweenness geometry, initiated by the Estonian scientists Sarv, Nuut, and
Humal in the 1930s, and by the author in 1964. It is established here that Tarski’s system of
geometry is essentially the same as Euclidean continuous betweenness geometry with a group
of movements.
Key words: Tarski’s system of geometry, betweenness geometry, group of movements.
1. INTRODUCTION
The recent paper [ 1 ] by Alfred Tarski (1902–83) and Steven Givant can be
considered as revival of Tarski’s system of geometry. Let us cite ([ 1 ], pp. 175, 176):
“In his 1926–27 lectures at the University of Warsaw, Alfred Tarski gave an
axiomatic development of elementary Euclidean geometry ... . [...] Substantial
simplifications in Tarski’s axiom system and the development of geometry based
on them were obtained by Tarski and his students during the period 1955–65. All
of these various results were described in Tarski [ 2 ¡ 4 ] and Gupta [ 5 ].” “[Section 2]
outlines the evolution of Tarski’s set of axioms from the original 1926–27 version
to the final versions used by Szmielew and Tarski in their unpublished manuscript
and by Schwabhäuser–Szmielew–Tarski [ 6 ].”
In Tarski’s system of axioms the only primitive geometrical objects are
points : a , b , c , ... . There are two primitive geometrical (that is non-
logical) notions: the ternary relation B of “soft betweenness” and quaternary
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relation ´ of “equidistance” or “congruence of segments”. The axioms are: the
reflexivity, transitivity, and identity axioms for equidistance; the axiom of segment
construction 9x ( B ( qax ) ^ax´bc )a.o.; reflexivity, symmetry, inner and outer
transitivity axioms for betweenness; the axiom of continuity, and some others.
In 1904, Veblen [ 7 ] initiated “betweenness geometry” with the same primitive
objects – points , and the only primitive notion – strict betweenness ; the name
betweenness geometry was given afterwards by Hashimoto [ 8 ].
This standpoint was developed further in Estonia, first by Nuut [ 9 ] in 1929 (for
dimension one, as a geometrical foundation of real numbers). In 1931 Sarv [ 10 ]
proposed a self-dependent axiomatics for the betweenness relation in the arbitrary
dimension n , extending Veblen’s approach so that all axioms of connection,
including also those concerning lines, planes, etc., became consequences. This
self-dependent axiomatics was simplified and then perfected by Nuut [ 11 ] and
Tudeberg (from 1936 Humal) [ 12 ]. As a result, an extremely simple axiomatics was
worked out for n -dimensional geometry using only two basic concepts: “point” and
“between”.
The author of the present paper developed in [ 13 ] a comprehensive theory
of betweenness geometry , based on this axiomatics (see also [ 14 ]). In [ 13 ] the
notions “collineation” and “flag” are defined in betweenness geometry, and also
the notion “group of collineations” is introduced by appropriate axioms. Using a
complementary axiom, this group is turned into the “group of motions”. These
axioms say that for two flags there exists one and only collineation in this group,
which transports one flag into the other.
The purpose of the present paper is to show that the axiomatics in [ 13 ] gives
the foundation of absolute geometry , the common part of Euclidean and non-
Euclidean hyperbolic (i.e. Lobachevski–Bolyai) geometry, and that by adding a
form of Euclid’s axiom, one obtains Tarski’s system of geometry.
2. TARSKI’S SYSTEM OF GEOMETRY
Recall that the original form of this system was constructed in 1926–27. It
appeared in [ 4 ], which was submitted for publication in 1940, but appeared only
in 1967 in a restricted number of copies. This paper (which is really a short
monograph) is reproduced on pp. 289–346 of Collected Papers [ 15 ], volume 4.
All the axioms are formulated in terms of two primitive notions, the ternary
relation of soft betweenness, B , and the quaternary relation of equidistance, ´ ,
among points of a geometrical space. The original set consists of 20 axioms for
2-dimensional Euclidean geometry. The possibility of modifying the dimension
axioms in order to obtain an axiom set for n -dimensional geometry is briefly
mentioned.
The next version of the axiom set appeared in [ 2 ]. A rather substantial
simplification of the axiom set was obtained in 1956–57 as a result of joint efforts
by Eva Kallin, Scott Taylor, and Tarski, and discussed by Tarski in his lecture
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course on the foundations of geometry given at the University of California,
Berkeley. It appeared in print in [ 3 ].
The last simplification obtained so far is due to Gupta [ 5 ]. It is shown in this
work that some axioms can be derived from the remaining axioms. As a result, the
axiom set consists of the following 11 axioms (see [ 1 ], also [ 6 ], 1.2):
Reflexivity , Transitivity , and Identity Axioms for Equidistance :
Ax.1. ab´ba ,
Ax.2. ( ab´pq ) ^ ( ab´rs ) !pq´rs ,
Ax.3 ab´cc!a = b .
Axiom of Segment Construction :
Ax.4 9x ( B ( qax ) ^ ( ax´bc )).
Five-Segment Axiom :
Ax.5 [( a6 = b ) ^B ( abc ) ^B ( a 0 b 0 c 0 ) ^ ( ab´a 0 b 0 ) ^ ( bc´b 0 c 0 )
^ ( ad´a 0 d 0 ) ^ ( bd´b 0 d 0 )] ! ( cd´c 0 d 0 ).
Inner Transitivity Axiom for Betweenness :
Ax.6 B ( abd ) ^B ( bcd ) !B ( abc ).
Inner Form of Pasch Axiom :
Ax.7 B ( apc ) ^B ( bqc ) !9x [ B ( qxa ) ^B ( pxb )].
Lowern-Dimensional Axiom forn =3 ; 4 ;::: :
Ax.8 ( n ) 9a9b9c9p 1 9p 2 :::9p n¡ 1
h V
1 ·i<j<n p i 6 = p j ^ V n¡ 1
i =2 ( bp 1 ´bp i ) ^ V n¡ 1
i =2 ( cp 1 ´cp i )
i
^ [ :B ( abc ) ^:B ( bca ) ^:B ( cab )]
:
Uppern-Dimensional Axiom forn =2 ; 3 ;::: :
Ax.9 ( n )
1 ·i<j<n p i 6 = p j ^ V n¡ 1
i =2 ( ap 1 ´ap i ) ^ V n¡ 1
i =2 ( bp 1 ´bp i ) ^ V n¡ 1
i
i =2 ( cp 1 ´cp i )
! [ B ( abc ) _B ( bca ) _B ( cab )] :
Axiom of Continuity :
Ax.10 9a8x8y [( x2X ) ^ ( y2Y ) !B ( axy )]
!9b8x8y [( x2X ) ^ ( y2Y ) !B ( xby )].
A Form of Euclid’s Axiom :
Ax.11 B ( adt ) ^B ( bdc ) ^ ( a6 = d ) !9x9y [ B ( abx ) ^B ( acy ) ^B ( ytx )].
This axiom set is denoted by EG ( n ) in [ 1 ]. In [ 6 ] the lower and upper dimension
axioms differ from those in EG ( n ) ; this axiom set is denoted by EH ( n ) .
In [ 1 ] it is noted that Tarski’s system of foundations of geometry has a
number of distinctive features, in which it differs from most, if not all, systems
of foundation of Euclidean geometry that are known from the literature. Of the
earlier systems, probably the two closest in spirit to the present one are those of
Pieri [ 16 ] and Veblen [ 7 ].
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i =2 ( ap 1 ´ap i ) ^ V n¡ 1
h V
3. CONTINUOUS BETWEENNESS GEOMETRY WITH MOVEMENTS
Veblen’s approach, based only on the axioms for betweenness, was developed
further in 1930–64 by the Estonian scholars Nuut, Sarv, Humal, and Lumiste
(see [ 9 ¡ 13 ]). The most complete studies, [ 10 ] and [ 13 ], were published in Estonian,
and are therefore not widely available. Betweenness geometry has been revived in
the author’s papers [ 17 ; 18 ].
The set of axioms for betweenness geometry is as follows [ 17 ]:
B1:( a6 = b ) )9c; ( abc );B2:( abc )=( cba );
B3:( abc ) ): ( acb );B4: habci^ [ abd ] ) [ cda ];B5:( a6 = b ) )9c;: [ abc ];
B6: : [ abc ] ^ ( abd ) ^ ( bec ) )9f; (( afc ) ^ ( def )) ;
where( abc )= B ( abc ) ^ ( a6 = b6 = c6 = a )means that b is strictly betweena
and c , habci =( abc ) _ ( bca ) _ ( cab )means that the triplet( a;b;c )is correct , and
[ abc ]= habci_ ( a = b ) _ ( b = c ) _ ( c = a )means that( a;b;c )is collinear .
Betweenness geometry as a system of consequences from these axioms is
developed in [ 10 ; 13 ; 14 ; 17 ; 18 ]. In [ 18 ] coordinates were introduced, algebraic
extension to ordered projective geometry was given, and collineations were
investigated.
In particular, the subset fxj ( axb ) g is called an interval ab with ends a and b . If
in an interval one has( axy ), then it is said that x precedes y . This turns the interval
into an ordered point-set.
If a;b;c are non-collinear, then they are said to be vertices , the intervals
bc;ca;absides (opposite to a;b;c , respectively) of the triangle4abc , which is
considered as the union of all of them.
In [ 13 ] it is proved (Theorem 13) that
For a triangle4abc , the subsetfxj9y; ( byc ) ^ ( axy ) gdoes not depend on the
reordering of verticesa;b;c .
It is natural to call this subset the interior of the triangle 4abc . Here any
permutation of a;b;c is admissible.
A betweenness geometry is said to be continuous (see [ 13 ]) if the following
axiom is satisfied.
Axiom of Continuity :
If the points of an intervalabare divided into two classes so that every point
xof the first class precedes every pointyof the second class , then there exists a
pointzwhich is either the last point of the first class , or the first point of the second
class ,
or, by means of only the betweenness relation (taking along also some concepts of
the set theory):
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B7: f ( ab = X[Y ) ^ [( x2X ) ^ ( y2Y ) ! ( axy )] g
!9f [( z2X ) ^ ( axz )] _ [( z2Y ) ^ ( azy )] g:
Among collineations movements can be introduced in betweenness geometry,
following [ 10 ] and [ 13 ].
First the configuration of a flag must be defined. Let us start with the line
L ab = fxj [ xab ] g , with a6 = b , the intervalab = fxj ( axb ) g , and the half-line
( ab = fxj ( axb ) _ ( x = b ) _ ( abx ) g of the line L ab , containing b , with initial point
a . In [ 13 ] and [ 17 ] it is proved that if two different points c;d belong to a line L ab ,
then L cd = L ab , i.e. the line is uniquely defined by any two of its different points.
Next the plane will be defined by P abc = Q a [Q b [Q c with non-collinear
a;b;c , where Q a = L ab [L ac S
x2bc L ax . It is obvious that the plane P abc does not
depend on the reordering of the points a;b;c . In [ 13 ] it is proved (Theorem 18) that
if three non-collinear points d;e;f belong to the plane P abc , then P def = P abc , i.e.
the plane is uniquely defined by any three of its non-collinear points. Moreover,
it is established (Theorem 19) that if two different points of a line L ab belong to a
plane P abc , then all points of this line belong to this plane.
It is said that two points x;y of the plane are on the same side of this line if
there is no point of this line between x and y , but they are on different sides of this
line if there exists z of this line so that( xzy ). In [ 13 ] it is proved (Theorem 22)
that a line L ab belonging to the plane decomposes all remaining points of this plane
into two classes so that every two points of the same class are on the same side of
this line, but every two points of different classes are on different sides of this line.
These classes are said to be half-planes , and this line is considered as their common
boundary . Here fxj9y [( y2L ab ) ^ ( cyx )]is the half-plane not containing c . The
other half-plane of P abc , with the same boundary L ab (i.e. containing c ), will be
denoted by( abjc .
A betweenness geometry is said to be two-dimensional (or plane geometry )
if there exist three non-collinear points a;b;c and all other points x belong to
P abc . For this geometry the flagF = F ( abc )is defined as the triple F ( abc )=
( a; ( ab; ( abjc ).
Any one-to-one map f of a betweenness plane onto itself is said to be a
collineation if( abc ) ) ( f ( a ) f ( b ) f ( c )), i.e. if the betweenness relation remains
valid by f . Then also[ abc ] ) [ f ( a ) f ( b ) f ( c )], i.e. every collinear point-triplet
maps into a collinear point-triplet. Hence every line maps into a line; from this
stems the term “collineation”. It is also clear that every flag maps into a flag, and
that all collineations of a betweenness plane form a group.
Now a new axiom for betweenness plane geometry can be formulated.
Axiom of Movements :
B8: For any two flagsFandF 0 there exists one and only one collineationfso
thatF 0 = f ( F ) .
In this situation f is called the movement which transports F into F 0 . It is clear
that all movements form a subgroup of the group of all collineations.
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