## Tarski's System of Geometry (1999)

Venue: | Bulletin of Symbolic Logic |

Citations: | 18 - 0 self |

### BibTeX

@ARTICLE{Tarski99tarski'ssystem,

author = {Alfred Tarski and Steven Givant},

title = {Tarski's System of Geometry},

journal = {Bulletin of Symbolic Logic},

year = {1999},

volume = {5},

pages = {175--214}

}

### OpenURL

### Abstract

. This paper is an edited form of a letter written by the two authors (in the name of Tarski) to Wolfram Schwabh auser around 1978. It contains extended remarks about Tarski's system of foundations for Euclidean geometry, in particular its distinctive features, its historical evolution, the history of specific axioms, the questions of independence of axioms and primitive notions, and versions of the system suitable for the development of 1-dimensional geometry. In his 1926--27 lectures at the University of Warsaw, Alfred Tarski gave an axiomatic development of elementary Euclidean geometry, the part of plane Euclidean geometry that is not based upon set-theoretical notions, or, in other words, the part that can be developed within the framework of first-order logic. He proved, around 1930, that his system of geometry admits elimination of quantifiers: every formula is provably equivalent (on the basis of the axioms) to a Boolean combination of basic formulas. From this theorem he...

### Citations

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Citation Context ...l real functions are Lebesgue-measurable can be proved, see Mycielski-sSwierczkowski [19] (where bibliographic references to earlier papers of Mycielski and Steinhaus can be found) as well as Solovay =-=[32]-=-. 17 It should be pointed out that the observations regarding the connections between the definability of B in terms of # and some properties of the underlying set theory apply also to the problem of ... |

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Citation Context ... In fact, we can give a precise description of all sets of reals which are definable in terms of + and 1 (such a description can be derived as a very particular consequence of the results in Szmielew =-=[36]-=-, but it can also be obtained in a much more direct way---see, e.g., the proof of Theorem 1 in Tarski [39], p. 232), and from this description we easily see that, e.g., the set of non-negative reals i... |

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Citation Context ... bisector of the segment zu, there is a point w on the perpendicular bisector of the segment xy whose distance from y is that same 15 An English translation this paper appeared as Article X in Tarski =-=[43]-=-. TARSKI'S SYSTEM OF GEOMETRY 203 . x . y . w . z . u . v Figure 23. The definition of # in terms of equidistance. as the distance from v to u---see Figure 23). It su#ces then to show that the followi... |

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Citation Context ...such a description can be derived as a very particular consequence of the results in Szmielew [36], but it can also be obtained in a much more direct way---see, e.g., the proof of Theorem 1 in Tarski =-=[39]-=-, p. 232), and from this description we easily see that, e.g., the set of non-negative reals is not so definable, although it is clearly definable in terms of + and #. As regards relations which can s... |

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Citation Context ...essarily more complicated. 13 We turn finally to Ax. 11. It is a modified form of the well-known Continuity Axiom which, in its application to the theory of real numbers, was first stated in Dedekind =-=[2]-=-. The modification consists in the removal of some additional conditions often imposed on X and Y such as the condition that X #Y is a straight line. This both simplifies and generalizes the formulati... |

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Citation Context ...an axiom even earlier, in Mollerup [16]. The Inner Pasch Axiom, Ax. 7, and Outer Pasch Axiom, Ax. 7 1 , are highly simplified and specialized variants of the well-known Pasch Axiom (Axiom IV in Pasch =-=[22]-=-; see also Axiom XXI in Pieri [23], mentioned above). In selecting Ax. 7 1 as an axiom in EG (n) and FG (n) , and deriving Ax. 7 as a theorem, Tarski followed the procedure in Schur [28], p. 7, where ... |

15 |
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Citation Context ...rem 5 di#er but very little from Ax. 7 1 and Ax. 7 respectively; on p. 9 of op. cit., Moore [17] is credited with the idea of this development. A weaker form of Ax. 7 1 occurs as Axiom VIII in Veblen =-=[50]-=-. The proof that this procedure can be reversed, i.e., that Ax. 7 can be selected as an axiom and Ax. 7 1 derived as a theorem, is due to Gupta [5], Theorem 3.70. We now turn to the lower and upper n-... |

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Citation Context ... with center z and radius zy, is if y = u---see Figure 24.) This presents no essential di#culties. The result is essentially due to Pieri [23]; in the outline above we follow the argument in Robinson =-=[25]-=-, pp. 70--71. As a consequence of the definability of B in terms of # we can adopt # as the only primitive notion of Euclidean geometry, and construct an axiom set for this geometry involving exclusiv... |

6 |
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Citation Context ...tary dimension-free geometry, i.e., the set of first-order sentences valid in all n-dimensional Euclidean geometries for n # 2. The conjecture that this is indeed the case is essentially due to Scott =-=[30]-=-, p. 66, and was a#rmed definitively by Gupta [5], p. 407. 8 Second, the sets EG (n) and EH (n) are complete, i.e., every first-order sentence, or its negation, is derivable from these sets. The axiom... |

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Citation Context ...lines 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 Schwabh auser-Szmielew-Tarski =-=[29]-=-. There follows, in Section 3, a discussion of the distinctive features of Tarski's approach to Euclidean geometry---the features which set it apart from other systems of Euclidean geometry that can b... |

4 |
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Citation Context ...tem 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 [41], [44], [45], and Gupta =-=[5]-=-. Aside from the importance of its metamathematical properties, Tarski's system of geometry merits attention because of the extreme elegance and simplicity of its set of axioms, especially in the fina... |

4 |
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Citation Context ...f a nonnegative number which, when added to x, yields y.) Regarding systems of set theory in which the fact that all real functions are Lebesgue-measurable can be proved, see Mycielski-sSwierczkowski =-=[19]-=- (where bibliographic references to earlier papers of Mycielski and Steinhaus can be found) as well as Solovay [32]. 17 It should be pointed out that the observations regarding the connections between... |

3 |
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Citation Context ... simple as, or even somewhat simpler than, Ax. 10 1 and have, perhaps, a somewhat clearer mathematical content. An example of such a sentence is Ax. 10 2 , which implies Ax. 10 1 by results of Bolyai =-=[1]-=- and which expresses the fact that every triangle can be inscribed in a circle. 11 Another example of such a sentence is Ax. 10 3 , which is somewhat longer than Ax. 10 1 , but---in opposition to the ... |

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Citation Context ... as the only primitive notion have been given for various important, finitely axiomatized subsystems of 2- and 3-dimensional elementary Euclidean geometry. See, for example, Schnabel [27], Grochowska =-=[4]-=-, and Richter-Schnabel [24]. 204 ALFRED TARSKI AND STEVEN GIVANT the fact that this ternary relation can be so used follows from results of Pieri [23]. There are various other quaternary and ternary r... |

2 |
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Citation Context ...ion to higher dimensional geometries, the problem of the definability of B in terms of # is rather involved. According to the observations of Lindenbaum announced (without proof) in Lindenbaum-Tarski =-=[9]-=-, the situation in the full system FG (1) can be described as follows. Using the Cartesian model we easily see that the problem discussed is equivalent to an analogous problem concerning the full syst... |

2 |
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Citation Context ... (and inessential) modification of Axiom XI in Veblen [51]. A somewhat more complicated but essentially equivalent form of Ax. 5, the Six-Segment Axiom, was used as an axiom even earlier, in Mollerup =-=[16]-=-. The Inner Pasch Axiom, Ax. 7, and Outer Pasch Axiom, Ax. 7 1 , are highly simplified and specialized variants of the well-known Pasch Axiom (Axiom IV in Pasch [22]; see also Axiom XXI in Pieri [23],... |

2 |
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Citation Context ...d deriving Ax. 7 as a theorem, Tarski followed the procedure in Schur [28], p. 7, where Postulate 6 and Theorem 5 di#er but very little from Ax. 7 1 and Ax. 7 respectively; on p. 9 of op. cit., Moore =-=[17]-=- is credited with the idea of this development. A weaker form of Ax. 7 1 occurs as Axiom VIII in Veblen [50]. The proof that this procedure can be reversed, i.e., that Ax. 7 can be selected as an axio... |

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Citation Context ...Euclidean geometry, for example the theory of n-dimensional Cartesian spaces over Pythagorean or Euclidean ordered fields, must always involve at least n + 2 distinct variables. For n # 3, Pambuccian =-=[20]-=- gives a set of axioms equivalent to CG (n) that uses exactly n + 2 distinct variables. The same paper gives sets of axioms equivalent to PG (2) and to CG (2) that use only six distinct variables and ... |

2 |
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Citation Context ...from most, if not all, systems of foundations 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 in Pieri =-=[23] and Veble-=-n [51]. One of the important features of Tarski's development is the clear distinction between the full geometry and its elementary part. By "elementary" we understand that portion of geomet... |

2 |
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Citation Context ...versal; it essentially expresses the fact that the sum of the angles of an arbitrary triangle is equal to two right angles. 12 The derivation of Ax. 10 1 from Ax. 10 3 is based on results of Saccheri =-=[26]-=-; see also Legendre [8]. It seems interesting that none of the above variants of the Parallel Axiom, Ax. 10--Ax. 10 3 , is directly related in its intuitive mathematical content to the basic idea of E... |

2 |
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Citation Context ...tion. Such sentences are useful if one desires to construct an axiom set for 2-dimensional a#ne Euclidean geometry, treating the betweenness relation as the only primitive notion; cf. Szczerba-Tarski =-=[34]-=- where Ax. 9 2 (2) is actually used for this purpose. 9 Ax. 9 2 (2) (and even Ax. 9 1 (2) ) is rather concise as compared with other known sentences which are formulated entirely in terms of the betwe... |

1 |
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Citation Context ...ch we shall not formulate here explicitly): as n increases, the sentences become really involved. Independent of the specific axioms discussed here it should be pointed out that, by results in Kordos =-=[6]-=-, no first-order universal-existential sentence formulated entirely in terms of betweenness can serve as a lower n-dimensional axiom for n # 3; hence, using dimensional axioms so formulated we can nev... |

1 |
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Citation Context ...lso some important gaps. [It is remarked on p. 26 of Schwabh auser-Szmielew-Tarski [29] that axioms Ax. 4, Ax. 5, Ax. 6, Ax. 8 (2) , Ax. 9 (2) , Ax. 10, and Ax. 11 are independent in FH (2) .] Kordos =-=[7]-=- states without proof the following information: in FH (2) , with Ax. 11 deleted and Ax. 10 replaced by Ax. 10 2 , the only independence problem that remains open is the one concerning Ax. 1. I do not... |

1 |
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Citation Context ...xpresses the fact that the sum of the angles of an arbitrary triangle is equal to two right angles. 12 The derivation of Ax. 10 1 from Ax. 10 3 is based on results of Saccheri [26]; see also Legendre =-=[8]-=-. It seems interesting that none of the above variants of the Parallel Axiom, Ax. 10--Ax. 10 3 , is directly related in its intuitive mathematical content to the basic idea of Euclid's original axiom,... |

1 |
Grundriss der reinen und angewandten Mathematik, vol
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(Show Context)
Citation Context ..., or its variant Ax. 10 1 in FG (n) , is one of the least known and most concise forms of the famous Axiom of Euclid (the Parallel Axiom). It is a specialized variant of the statement found in Lorenz =-=[10]-=-, vol. I, pp. 101--102, to the e#ect that every interior point of an angle lies always on a line intersecting the two sides of the angle. 10 As in the case of Ax. 9 1 (2) , it is formulated entirely i... |

1 |
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Citation Context ... dimensions, we have to change our conception of 1-dimensional geometry and provide it with stronger primitive notions. As an example we may consider the quaternary relation K discussed in Makowiecka =-=[11]-=-, [12]. In the field of real numbers it is defined by the elementary formula K(xyzu) # [(x - u) 2 = (y - u) (z - u)]. The relation K is a geometric notion in the sense that it is preserved under every... |

1 |
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Citation Context ...articular real numbers elementarily definable in terms of +, # and 1 are rational numbers; however the irrational number # 2 is obviously elementarily definable in terms of + and (see also Makowiecka =-=[13]-=-, pp. 677--678). Makowiecka has given several other examples of geometric relations which are elementarily interdefinable with K in 1-dimensional space, so that each of them can serve as the only prim... |

1 | Euklidische Raume - Richter, Schnabel - 1997 |

1 |
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Citation Context ...distance relation as the only primitive notion have been given for various important, finitely axiomatized subsystems of 2- and 3-dimensional elementary Euclidean geometry. See, for example, Schnabel =-=[27]-=-, Grochowska [4], and Richter-Schnabel [24]. 204 ALFRED TARSKI AND STEVEN GIVANT the fact that this ternary relation can be so used follows from results of Pieri [23]. There are various other quaterna... |

1 |
nski, Cardinal and ordinal numbers, Pa nstwowe Wydawnictwo Naukowe
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Citation Context ...ther increasing nor decreasing. As is well-known, with the help of the axiom of choice we can establish the existence of such a function g and show, without using the axiom of choice (see Sierpisnski =-=[31]-=-, p. 442), that no such function is measurable. From properties of g it follows easily that g(1) #= 0. Hence we can define a new function f by stipulating that f(x) = g(x)/g(1) for every x # R. Of cou... |

1 |
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Citation Context ...dependence of the three remaining axioms Ax. 1, Ax. 3, and Ax. 7 1 is still an open question. If, on the other hand, we modify FG (2) by replacing Ax. 10 1 with Ax. 10 2 , then, according to Szczerba =-=[33]-=-, axiom Ax. 7 1 proves to be independent as well. To our knowledge, the problem whether Ax. 7 1 is independent in FG (2) itself remains open. This problem seems to be connected with the problems conce... |

1 |
discussion of some a#ne geometries
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Citation Context ...ss relation and can serve as upper 2-dimensional axioms. The situation changes essentially if we take 9 The full version of this paper appeared soon after these notes were written; see SzczerbaTarski =-=[35]-=-. TARSKI'S SYSTEM OF GEOMETRY 197 up the task of constructing n-dimensional analogues, Ax. 8 2 (n) and Ax. 9 2 (n) , of Ax. 8 (2) and Ax. 9 2 (2) (which we shall not formulate here explicitly): as n i... |

1 |
a#ne to Euclidean geometry: An axiomatic approach, translated by M. Moszy nska, PWN-Polish Scientific
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Citation Context ...tten. However, the project was never completed. Over the years, Szmielew had gradually changed her views on the foundations of geometry, and had begun a development along di#erent lines (see Szmielew =-=[38]-=- and the review Moszysnska [18]). Her untimely death put an end to all prospects for completion of the work. Eventually, Wolfram Schwabh auser did prepare such a treatise (in German) , based in part o... |

1 |
is elementary geometry?, The axiomatic method, with special reference to geometry and physics
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Citation Context ... 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 [41], =-=[44]-=-, [45], and Gupta [5]. Aside from the importance of its metamathematical properties, Tarski's system of geometry merits attention because of the extreme elegance and simplicity of its set of axioms, e... |

1 |
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(Show Context)
Citation Context ...ensions is lacking. As was pointed 20 Recall that these words were written around 1978. 21 I [Tarski] made some search, both in my memory and in the NSF project reports, Tarski et al. [48], [49], and =-=[47]-=-. I reached the following conclusions concerning the chronology of Wanda's work on the manuscripts involved; they are not certain, but seem to me very plausible. Some time during Wanda's stay in Berke... |