tarski and geometry

8/2/2019 Tarski and Geometry

1/7

Tarski and GeometryAuthor(s): L. W. SzczerbaSource: The Journal of Symbolic Logic, Vol. 51, No. 4 (Dec., 1986), pp. 907-912Published by: Association for Symbolic LogicStable URL: http://www.jstor.org/stable/2273904

Accessed: 08/01/2010 09:41

Your use of the JSTOR archive indicates your acceptance of JSTOR's Terms and Conditions of Use, available at

http://www.jstor.org/page/info/about/policies/terms.jsp. JSTOR's Terms and Conditions of Use provides, in part, that unless

you have obtained prior permission, you may not download an entire issue of a journal or multiple copies of articles, and you

may use content in the JSTOR archive only for your personal, non-commercial use.

Please contact the publisher regarding any further use of this work. Publisher contact information may be obtained at

http://www.jstor.org/action/showPublisher?publisherCode=asl.

Each copy of any part of a JSTOR transmission must contain the same copyright notice that appears on the screen or printed

page of such transmission.

JSTOR is a not-for-profit service that helps scholars, researchers, and students discover, use, and build upon a wide range of

content in a trusted digital archive. We use information technology and tools to increase productivity and facilitate new forms

of scholarship. For more information about JSTOR, please contact [email protected].

Association for Symbolic Logic is collaborating with JSTOR to digitize, preserve and extend access to The

Journal of Symbolic Logic.

http://www.jstor.org
http://www.jstor.org/stable/2273904?origin=JSTOR-pdfhttp://www.jstor.org/page/info/about/policies/terms.jsphttp://www.jstor.org/action/showPublisher?publisherCode=aslhttp://www.jstor.org/action/showPublisher?publisherCode=aslhttp://www.jstor.org/page/info/about/policies/terms.jsphttp://www.jstor.org/stable/2273904?origin=JSTOR-pdf


2/7

THE JOURNALOF SYMBOLICLOGICVolume 51, Number 4, Dec. 1986

TARSKI AND GEOMETRY

L. W. SZCZERBA

Tarski published his first geometry paper, [24b], in 1924. As is well known, thearea of the union of two disjoint figuresis the sum of the areas of these two figures.This observation is the basis of a method for proving that two figures,say A and B,have the same area: if we can divide each of the two figures A and B into a finitenumberof pairwisedisjoint subfiguresA1,-.. , A,,and B1,. . ., B, such that forevery i,figuresAi and Biare congruent (we say that two such figuresareequivalentby finitedecomposition),hen figuresA and B have the same area.The method is by no meansuniversal. For example a disc and a rectangle can never be equivalent by finitedecomposition, even if they have the same area. Hilbert [1922, Kapitel IV] provedfrom his axiom system the so-called De Zolt axiom:If a polygon V is a propersubset 6f a polygon W then they are not equivalentby afinite decomposition.Hilbert's proof is elementary but difficult. In [24b] Tarski gave an easy butnonelementary proof of a stronger version of the De Zolt axiom:If a polygon V is a propersubset of a polygon W then they are not equivalentbyfinite decomposition nto any figures.Moreover,he pointed out that the above theoremsaretrue only on the plane.Anytwo polyhedraareequivalentby finitedecomposition (cf.Banach and Tarski [24d]),even if they differ in volume! This paper contains the famous Banach-Tarskiparadox, an unusual consequence of the axiom of choice.If two polygons, say V and W, are equivalent by finite decomposition intopolygons it is natural to ask what is the minimal number of polygons in such apartition. Such a numberis called the degreeof equivalenceof V and W. The notionis due to A. Lindenbaumand is describedin Tarski [31b]. Let Vbe a squarewith thelength of a side equal to a, and let W be a rectangle with sides of length ax and a/xrespectively (where x is any positive real number). The square and rectangle areeasily seen to be equivalent by finite decomposition. The degree of equivalence isdenoted by z(x). The problem of propertiesof the function Tinterested a numberofwell-known young mathematicians in the 1920s, among them Knaster, Lin-denbaum, Moese and Waraszkiewicz.The results of their efforts were reported in

Received April 22, 1985; revised January 20, 1986.( 1986, Association for Symbolic Logic0022-4812/86/5104-0006/$0 1.60

907


3/7

908 L. W. SZCZERBATarski [31/32a], including Moese's theorem that for all natural numbers n we haveT(n) = n.Later W. Sierpifiski published a comprehensive survey of the problem ofequivalence by finite partitions [1954].During the academicyear 1926/27 Tarskigave a course on Euclidean geometry atthe University of Warsaw.For those unfamiliarwith the Polish university system Iowe an explanation: courses in Polish universitiesareusually held for two hours perweek, but are spread over a year or a semester.Tarski's geometry course took twosemesters:Fall semester of 1926 and Spring semester of 1927.The system that Tarski presented in this course was designed after Pieri [1908](rather than Hilbert [1922]) and contained a number of innovations. Only oneuniverse was used, the set of points, with two undefined (primitive) relations:betweenness and equidistance. Tarski strongly opposed the practiceof formulatingaxioms with the use of defined notions. By 1929 Tarski was able to prove that hissystem was complete in the sense that every sentence formulated in terms ofprimitive notions was either provable or disprovable, and that there exists amechanical method of finding these proofs.The axiom system used in this course was about to be published in Francein 1940.It was already in press, but because of the outbreak of the Second World War itfailedto come out. Tarski got J.C. C. McKinsey to write it once more and publishedit in [48m]. Surprisingly, in 1967 on his 65th birthday Tarski was able to see theoriginal paper reproduced [67ma] from galley proofs which had fortunately beenfound.A modification of this axiom system was published in Tarski [59], wherethe termelementarygeometrywas explained as follows: "We regardas elementarythat partofEuclidean geometry which can be formulated and established without the help ofany set-theoretic devices." Thus Tarski insisted that the axiom system should beof the first order, i.e. no variables for sets should be used. The main difficultycon-sisted in providing an analog of the continuity axiom. Tarski took the Dedekindform of the continuity axiom

VAVB(A < B = 3xA < x < B)as a startingpoint. Firsthe had to expressit in termsof his primitives(in this case thebetweenness relation suffices):

VAVB(3xVa,b(a E A & b E B =* Bxab)) => (3yVa,b(a E A & b E B > Bayb)).The second step was to eliminate set variables. It was achieved by restricting theconsiderations to definablesets. If A and B in the above axioms rangeover definablesets, then it is possible to replacethem by definitions of these sets. Thus he arrivedatthe collection of sentences All (see below). The resulting axiom system iselementary, but contains an infinite number of axioms. This disadvantage of theaxiom system is unavoidable: Elementary Euclidean geometry is not finitelyaxiomatisable.Tarski had, for a long time, planned to write a monograph on Euclideangeometry, and in the early 1960s he began to realize this plan in collaboration withWanda Szmielew. A first draft was completed in 1965 and the second in 1967. For


4/7

TARSKI AND GEOMETRY 909various reasons a complete monograph was never published. In [83'] Schwab-hauser presented a detailed development of the Tarski axiom system, based in parton the 1965 notes of Wanda Szmielew. The first part of the book [83'] containingthe exposition was published under three names: Schwabhiuser, Szmielew andTarski.Szmielewdid not live to see its publication. Tarski died a few months after itcame off the press. The second part, containing a comprehensive description ofmetamathematical investigations of Tarski's system of geometry and "Tarski style"metamathematical results on a number of related systems, was published under thename of Schwabhauseralone.The axiom system given in [83'] is the following (ab cd is read "a is the samedistance from b as c is from d",and Babc is read "bis between a and c"):Al. ab ba [reflexivity].A2. ab pq A ab rs =. pq _ rs [transitivity].

A3. ab cc => a = b [identity for equidistance].A4. 3x(Bqax A ax _ bc) [segment construction].A5. a : b A Babc A Ba'b'c' A ab a'b' A bc b'c' A bd -b'd' => cd c'd'[five-segment axiom].A6. Baba < a = b [identity for betweenness].A7. Bapc A Bbqc =* 3x(Bpxb A Bqxa) [axiom of Pasch].A8. 3a0b3c(n Babc A - Bbca A - Bcab) [lower dimension axiom].A9. p :# q A ap _ aq A bp bq A cp -cq => Babc v Bbca v Bcab [upper di-mension axiom].A10. Badt A Bbdc A a :# d 3x3y(Babx A Bacy A Bxty) [Euclid's axiom].All. 3aVxVy (O(x) A (y) Baxy) =- 3bVxVy (4(x) A 0(y) =* Bxby) [Con-tinuity axiom-schema].In the last axiom q and , are first order formulas in the appropriate language,such that the variablesa, b,x do not occur in f, and the variables a, b, y do not occurin q.The approach to geometry underlying the above exposition is axiomatic.Therefore it is natural to start with axioms Al -A9 and All forming an axiomsystem of absolute geometry, and to extend it later with A10 to form Euclidean

geometry, or with the negation of A1O o obtain Bolyai-Lobachevsky geometry. Ananalog of Tarski [59] for Bolyai-Lobachevsky geometry was published by Szmielew[ 1959]. In fact the main partof the development of Euclideangeometry does not useAl 1,and this leads to a broad class of underlyingfields. It is the class of all formallyreal Pythagorean fields, i.e. fields in which a sum of squares is always a square,but- 1 is not a sum of squares. Only the addition of A 1Ilimits the class to the class ofreal-closed fields. Tarski was unhappy that he was unable to repeat the same forBolyai-Lobachevsky geometry: the Klein model makes good sense only over aformallyreal Euclideanfield,i.e. a field in which forany element x eitherx or - x is asquare. This idea has been realized by Szmielew in her monograph [1983], editedposthumously by M. Moszyfiska, in fact starting with a much broader class ofunderlying algebraic systems, but at the expense of giving up Bolyai-Lobachevskygeometry.Tarski insisted that the exposition should be formal and no recourse to intuition,and in particularto figures, should be made. Nevertheless he always stressed that the


5/7

910 L. W. SZCZERBAexposition should be easy and appealing to intuition. I remember Szmielewcomplaining jokingly that the more one worked with Tarski, the result tended tolook less and less laborious. In fact Tarski would work over a mathematicalpresentation until it achieved an elegance and simplicity which disguised thedifficulties hidden beneath the surface. All these requirements applied also toaxioms, and thereforeTarski wanted his axiom system to be independent. Most ofthe independence proofs have been provided by Eva Kallin, Scott Taylor andH. N. Gupta (see Gupta [1965]). Gupta left the problem open for just two axioms,Al and A7, and this is the state of affairsnow (Szczerba [1970] contains the proof ofindependence of A7 but in a system with another version of axiom A10; cf. alsoSzczerba and Szmielew [1970]).Besides Euclidean geometry Tarski was interested in general affine geometry,which is, roughly speaking, the theory of the betweenness relation on the open andconvex subsets of the Euclidean plane (for a precise definition see Szczerba andTarski [65a] and [79]). Tarski was especially interested in the rich variety ofextensions of this theory. Among these extensions are Euclidean affine geometry,hyperbolic and elliptic geometries and a number of less natural items (fordetails see[79]). A considerable part of both papersis devoted to the study of metamathemat-ical properties of general affinegeometry and its extensions.Studying metamathematical properties of particular theories was quite charac-teristic for Tarski, and geometry was one of his major sources of examples. Forexample, he solved the decidability problem for projective geometry (see [49ai]), andwrote a number of papers about primitive notions in geometry. The problem ofprimitive notions in geometry seems to be particularly prominent in Tarski'sgeometric research. As a disadvantage of Hilbert's axiom system for Euclideangeometry Tarski mentioned the fact that Hilbert used a number of mutuallydefinable primitive notions, and he was unhappy that he himself succeeded inimproving the Hilbert system of primitivenotions only up to the point wherehe hadjust one notion definable from the other: the betweenness relation is definable interms of equidistance in Tarski's system of Euclidean geometry. This was the pricehe paid for having a nice axiom system, but he often stressed the need forimprovement and looked for another system of primitives. He would gladly havehad an axiom system with one relationalprimitiveonly. (The betweennessrelation isdefinablein terms of equidistance provided the dimension is at least two. TarskiandLindenbaum proved in [26aa] that in the one-dimensional case these two notionsare definitionally independent.)The search for such a system resulted in the quite general theorem that ifequidistance on the Gaussian plane is definable from a ternary relation R then forany two different complex numbers a and b the set {c:a = c v b = c v Rabc}generates the entire field of complex numbers (see Tarski [56c]). Particularresultsalong this line are two: Together with Beth,he proved that equilateralitymay be thesole primitiveof Euclideangeometry (see [56b]) but only fordimension at least 3. Indimension 2 the equilaterality does not satisfy the condition from [56c]. Theproblem of possible primitives of Euclidean geometry has been undertaken byRoyden [1959], Scott [1956], and Makowiecka in a series of articles (see e.g.[1977]).


6/7

TARSKIAND GEOMETRY 911It is not necessary to confine oneself in choosing primitives to points, lines, etc. asthe elements of universa.Why not take, say, balls?It turns out (see Tarski [29]) thatin terms of open balls (or discs on the plane) and their set-theoretical inclusion, it ispossible to define points and equidistance in any finite-dimensional Euclidean

space. In fact Tarski did not use the relation of inclusion but instead formulatedhisresult in the framework of Le'niewski's mereology (cf. Luschei [1962] for adescription) based on the notion of part.This system of primitives,i.e. open balls and inclusion, was motivated not only byits simplicity, but also by the fact that the notion of ball seems to be much moreintuitive than the notion of point. Thus the theory based on these notions hasbecome known as Tarski's natural geometry. This natural geometry differssignificantly from similar approaches of Pieri [1908] and Huntington [1916].Jaskowski in [1948] simplified Tarski's natural geometry by taking closed ballsinstead of open ones.It is impossible to describein a short paper all the researchin geometry which hasbeen influencedby Tarski. Instead I will referthe readerto Szmielew [1974] and thesecond part of [83m]. Both contain comprehensive bibliographies.This paper is an extended version of a talk presentedat the meeting of the PolishPhilosophical Society on the first anniversary of Tarski's death. I would like toexpress my gratitude to Andrzej M4kowski, Steven Givant and the anonymousreferee for their remarks leading to the improvement of the final version of thepaper.

REFERENCES(OTHER THAN WORKS OF TARSKI)

H. N. GUPTA[1965] Contributions to the axiomatic foundations of geometry. Ph.D. Thesis, University of

California, Berkeley, California.D. HILBERT

[1922] Grundlagen der Geometrie, 3rd ed., Teubner, Leipzig.E. V. HUNTINGTON

[1916] A set of postulates for abstract geometry exposed in terms of the simple relation of inclusion,Mathematische Annalen, vol. 73, pp. 522-559.

S. JA?KOWSKI[1948] Une modification des definitions foundamentales de la gdomdtrie des corps de M. A. Tarski,

Annales de la SocietW Polonaise de Mathimatique, vol. 21, pp. 298-301.E. C. LUsCHEI

[1962] The logical systems of Lesiniewski, North-Holland, Amsterdam.H. MAKOWIECKA

[1977] On minimal systems of primitives in elementary Euclidean geometry, Bulletin de l'AcadimiePolonaise des Sciences, Sirie des Sciences Mathimatiques, Astronomiques et Physiques, vol. 13,pp. 269-277.

M. PIERI[1908] La geometria elementare instituita sulle nozione di "punto" e "sfera", Memorie di Matematica

e di Fisica delta Societa Italiana delle Scienze, ser. 3, vol. 15, pp. 345-450.H. L. ROYDEN

[1959] Remarks on primitive notions for elementary Euclidean and non-Euclidean plane geometry, Theaxiomatic method (proceedings of the 1957/58 international symposium, Berkeley, California;L. Henkin et al., editors), North-Holland, Amsterdam, pp. 86-96.


7/7

912 L. W. SZCZERBAD. SCOTT[1956] A symmetricprimitivenotion for Euclideangeometry,IndagationesMathematicae,vol. 18, pp.456-461.W. SIERPI&SKI[1954] On the congruenceof sets and theirequivalenceby finite decomposition,Lucknow University

Studies,no. 20, Lucknow University, Lucknow, 1954;reprinted n Congruenceof sets andothermonographs,Chelsea, New York, 1967.L. W. SZCZERBA[1970] Independenceof Pasch's axiom, Bulletin de l'Acadimie Polonaise des Sciences, Sirie desSciences Mathematiques,Astronomiqueset Physiques,vol. 18, pp. 491-498.L. W. SZCZERBA and W. SZMIELEW[1970] On the Euclidean geometry without the Pasch axiom, Bulletin de l'Acadimie Polonaise desSciences, Sirie des SciencesMathematiques,Astronomiqueset Physiques,vol. 18,pp. 659-666.W. SZMIELEW

[1959] Some metamathematicalproblemsconcerningelementary hyperbolicgeometry, The axiomaticmethod (proceedingsof the 1957/58 internationalsymposium,Berkeley, California;L. Henkinet al., editors), North-Holland, Amsterdam,pp. 30-32.[1974] The role of the Pasch axiom in the foundations of Euclideangeometry, Proceedings of theTarski symposium(L. Henkin et al., editors), Proceedings of Symposia in Pure Mathematics,vol. 25, American Mathematical Society, Providence, Rhode Island, pp. 123-132.[1983] From affine to Euclidean geometry, Panistwowe Wydawnictwo Naukowe, Warsaw, andReidel,Dordrecht.INSTITUTE OF MATHEMATICS

WARSAW UNIVERSITYWARSAW, POLAND

tarski and geometry

Documents