A General Theorem Concerning Primitive Notions of Euclidean Geometry

MATHEMATICS

A GENERAL THEOREM CONCERNING PRIMITIVE NOTIONS OF EUCLIDEAN GEOMETRY BY

ALFRED TARSKI (Communicated by Prof. A.

HEYTING

at the meeting of May 26, 1956)

In this note I should like to generalize certain observations made in the joint article of E. W. BETH and myself, [1]. The observations concerned some particular ternary relations between points which can, or cannot, be used as the only primitive notions in developing Euclidean geometry. I shall establish here a common property of all those ternary relations which can serve as unique primitive· notions for Euclidean geometry. The result may be of some value for the reason that it can be readily applied to special relations; in various particular cases it is easy to show that a given relation does not have the property established in the general result. With some changes in formulation, the result extends to relations with more than three terms and to systems of relations. 1 ) Whether a given relation can be used as the only primitive notion in n-dimensional geometry sometimes depends on the value of n. For instance, among relations discussed in [l ], PIERI's relation I can be used this way for n=2, 3, ... ,but cannot be used for n=l; by Theorem l of [l ], the relationE can be used this way for n = 3, 4, ... while, by Theorem 2, it cannot be used for n= l, 2. As an example ~fa ternary relation between points which can be used as the only primitive notion for all values of n we may mention the relation J which holds between three points x, y, and z if and only if the distance from x to y at most equals the distance from x to z. In case n > 2 this follows from the fact that the relation I is definable in terms of J since the formula I(x, y, z) is equivalent to the conjunction of formula J(x, y, z) and J(x, z, y). In case n= l we notice that, in addition to I, the betweCimess relation B (which holds between three points x, y, and z if and only if y lies on the segment connecting x and z) is definable in terms of J since, in this case, B(x, y, z) is equivalent to the conjunction of J(x, y, z) and J(z, y, x); on the other hand, it is known that all geometrical concepts are definable in terms of I and B. Consequently, a relation (or any other notion) R can then and only then serve as the only primitive notion for n-dimensional Euclidean geometry when, within this geometry, R is inter-definable with J (i.e., when either of these two relations is definable in terms of the other). 1) I am indebted to Mr. draft of this note.

DANA ScoTT

for pointing out a flaw in the original

469 The concept of definability acquires a precise meaning only if it is referred to a well-determined formalism and if therefore all the admissible forms and means of definition are exactly specified. For our purposes,. however, the way in which Euclidean geometry has been formalized is rather irrelevant. The reason is that in our discussion we shall use only one property of the concept of definability, and in fact a property which holds in all known formalisms. It can be formulated as follows: (I) If the notion N is definable in terms of the notions Nv ... , N m within a given theory, then N is invariant under every one-to-one transformation of the universe of discourse of this theory onto itself under which all the notions Nv ... , N m are invariant. 2 )

In particular, it is obvious that the ternary relation J between points of the n-dimensional Euclidean space S is invariant under every similarity transformation of this space, i.e., under every one-to-one transformation T of S onto itself satisfying the following condition: there is a positive real number r such that

d(Tx, Ty) =.r·d(x, y) for any points x andy of S (where d(x, y) denotes the distance from x toy). It can also be shown that, conversely, every one-to-one transformation of S onto itself under which J is invariant is a similarity transformation. For n = 1 this property of J can easily be established in a direct way; if n;> 2, we notice that the property applies to the relation J, as a result of the discussion in [3] (cf. pp. 389 and 442), and that consequently it applies to J since I is definable in terms of J. In view of (I) both these properties of J automatically extend to every relation (and, more generally, to every notion) R which is inter-definable with J. We thus arrive at the following statement, which forms the base for our subsequent discussion: 2) In this formulation (I) applies to interpreted theories, i.e., to theories which are provided with definite interpretations of symbols occurring in them; moreover, each such theory is assumed to have a well determined universe of discourse -a set U such that all the notions of the theories are intrinsic with respect to U (i.e., they are subsets of U or relations between elements of U or relations between these subsets and relations, etc.). The formulation of (I) must be modified if (I) is to apply to non-interpreted theories. In this case we speak of the definability of a constant 0 in terms of other constants 0 1 , ... , Om; we consider all the models of a given theory-each such model ill1 is formed by a set U(ill't) and by certain notions N(ill't), N 1 (ill1), ... , Nm(ill't), ... which are intrinsic with respect to U(ill't) and which (because of their logical structure) can serve as interpretations of the constants 0, 01, ... , om, .... If now 0 is definable in terms of 01, ... , om within a theory T, then, in every model ill1 ofT, the notion N(ill't) is invariant under every one-to-one transformation of the set U(ill't) onto itself under which all the notions ·N1(IDl), ... , NmODl) are invariant. In connection with (I) compare [3], in particular Theorem 6 and the concluding remarks of Section l.

470

(II) If a relation R between points of the n-dimensional space S can serve as the only primitive notion for n-dimensional Euclidean geometry, then the set of all one-to-one transformations of S · onto itself under which R is invariant coincides with the set of all similarity transformations of S. We can now formulate and establish the main result of this note. To avoid some technical complications we formulate it explicitly only for the case of two dimensions: Theorem l. Let R be a ternary relation between points of the Euclidean plane (identified with complex numbers) which can serve as the only primitive notion for two-dimensional Euclidean geometry. Let a 1 and a 2 be any two distinct complex numbers, and 0 be the set of all complex numbers c for which R(av a 2 , c) holds. Then the number field 0 generated by the set consisting of a 1 , a 2 , and all the elements of 0 contains all complex numbers. Proof: As is well known, with any two distinct complex numbers x and y we can correlate the similarity transformation T x. v analytically expressed by the formula

where u is an arbitrary complex number; clearly,

Since, by hypothesis and in view of (II), R is invariant under every similarity transformation, we easily obtain the following conclusion: (l) for any complex numbers x, y, and z with xi=y we have R(x, y, z) if and only if

for some element c of 0. We see thus that, apart from the triples x, y, z with x=y, the set 0 uniquely determines the relation R. We now proceed essentially as in the proof of Theorem 2 in [1]: we treat the set of complex numbers as a vector space over the number field 0 and, using in general the axiom of choice, we construct a basis for this space, i.e., a set B of complex numbers satisfying the following condition: (2) every complex number x can be uniquely represented in the form X

=

C1 ·

b1 + ... + Cm • bm

where Cv ..• , em are any elements of 0 different from 0 while bv ... , bm are any elements of B. Moreover, we assume that B contains the number l. In view of (2) we can define a function T on and to the set of complex numbers by means of the following stipulations:

471

(3) T1 = 2; Tb = b for every element b of B which is different from 1; (4) if X = !1_ • b1 + ... +em •bm is the representation of a complex number x described in (2), then

Tx

= !1_ • Tb1 +

... +em· Tbm.

From (2)-(4) we derive the following properties ofT: (5) T is a one-to-one transformation of the set of all complex numbers onto itself; (6) T(x+y)=Tx+Ty for any complex numbers x and y; (7) T(e·x)=e·Tx for every element e of 0 and every complex number x. Next we show that (8) for any complex numbers x, y, and z the formulas R(x, y, z) and R(Tx, Ty, Tz) are equivalent. In case x-=Fy this readily follows from (1) and (5)-(7), since, for every number e of 0, the numbers az-C

and

c-al

obviously belong to 0. In case x=y the ternary relation R reduces to a binary relation between x and z expressed by the formula R(x, x, z). As is known, there are only four binary relations which are invariant under every similarity transformation: R(x, x, z) holds either for any two points x and z or for no points or for those and only those points which are identical or, finally, for those and only those points which are different. Using (5) we easily check that (8) is satisfied in each of these four cases. By (5) and (8), Tis a one-to-one transformation of the plane onto itself under which R is invariant. Hence, by (II) and the hypothesis, T is a similarity transformation. Suppose now, contrary to the conclusion of our theorem, that the number field 0 does not contain all complex numbers. From (2) it follows then that the basis B contains at least two different elements and hence an element b' -=F 1; we also see from (2) that 0 does not belong to B. By (7), with e=O, and (3) we have

TO=O, T1 =2, and Tb' =b' with b' -=FO; these formulas clearly imply that T is not a similarity transformation. We have thus arrived at a contradiction, and therefore we have to assume that 0 contains all complex numbers. This completes the proof. From the proof of Theorem 1 it is seen that the theorem remains valid if 0 is defined as the number field generated by the numbers

472

where c ranges over all the elements of 0; in case a 1 = 0 and a2 = l, 0 is simply the number field generated by C. It is also seen from this proof that the power of the set 0 does not depend on the choice of the numbers a 1 and a 2 (provided a 1 c#a2 ). If, for some particular couple of these numbers, the set 0 has only finitely many or denumerably many or, more generally, · less than continuum elements, then the number field 0 has also less than continuum elements and hence cannot contain all complex numbers. Thus as a direct consequence of Theorem 1 we obtain Corollary 2. Under the hypothesis of Theorem 1, the set 0 has the power of the continuum. Both Theorem 1 and Corollary 2 extend to Euclidean geometry with an arbitrary number n of dimensions. For n = 1 we simply replace in Theorem 1 complex numbers by real numbers and the Euclidean plane by the Euclidean straight line; for n > 3 the formulation of Theorem 1 becomes more involved. On the other hand, Corollary 2 (with trivial changes) applies to any number of dimensions. Theorem 1 and Corollary 2 can also be extended to any systems of ternary relations Rv ... , Rm which can serve as the full system of primitive notions for n-dimensional Euclidean geometry; in case n = 2, 0 is defined as the set of all complex numbers c for which at least one of the formulas R 1 (a1 , a 2 , c), ... , Rm(av a 2 , c) holds. Finally, we can extend these results to relations with more than three terms, e.g., to a quaternary relation R; 0 is then defined as the set of all numbers c such that, for some number d, at least one of the formulas R(av a 2 , c, d) and R(~, a 2 , d, c) holds. To conclude, we want to give some applications of our general results to special relations. (i) Let E be the relation discussed in. Theorems 1 and 2 of [1 ], i.e., the relation which holds between three points x, y, and z if and only if these points are the vertices of an equilateral triangle or else if they coincide. In case n = 2, for any two distinct points x and y there are only two points z for which E(x, y, z) holds; in case n= 1, such points z do not exist at all. Hence for the relation R = E the set 0 described in the hypothesis of our Theorem 1 is finite, and therefore, by Corollary 2, E cannot serve as the only primitive notion for one- or two-dimensional Euclidean geometry. Thus Theorem 2 of [1] can be derived as a particular case from Corollary 2. The same argument applies to the ternary relation R and the quaternary relation Q mentioned at the end of [1 ]. (ii) Consider Pieri's relation I. In case n= 1, for any two distinct points x and y there are only two points z with I(x, y, z). Hence, by Corollary 2 (applied to the case n= 1), the relation I cannot serve as the only primitive notion for one~dimensional geometry; as was stated in [1], this result was first obtained by LINDENBAUM.

473

(iii) Let K be the ternary relation which holds between points x, y and z of the Euclidean plane if and only if z can be constructed from x and y by means of a ruler and a pair of comp~sses. For any two distinct points x and y there are only denumerably many points z such that K(x, y, z) holds. Hence, again by Corollary 2, K cannot serve as the only primitive notion for two-dimensional geometry. (iv) Let L be the ternary relation which holds between points x, y, and z of the plane if and only if they are collinear. In this case the set C of our Theorem 1 coincides with the set of all real numbers, and so does the number field 0 generated by C. Thus 0 does not contain all complex numbers and therefore, by Theorem 1, L cannot serve as the only primitive notion for two-dimensional Euclidean geometry. The same conclusion holds for every number n > 2 of dimensions (but it requires an appropriate extension of Theorem 1). Furthermore, the conclusion extends to any three- or more-termed relation R and any system of such relations Rv ... , Rm, provided the points x, y, z, ... between which the relation R ol' each of the relations R; holds are always collinear; thus, for instance, to the betweenness relation B mentioned in an earlier part of our discussion. 3 ) This general conclusion trivially does not apply to the case n= l. On the other hand, it can be shown directly that, e.g., the relation B cannot serve as the only primitive notion for one-dimensional Euclidean geometry; but we see no possibility of deriving this conclusion from our general results. It may be mentioned that the axiom of choice plays an essential role in establishing·our general results as well as in applications (i), (ii), (iii). 4 ) However, application (iv) is independent of this axiom (since in this particular case the basis B in the proof of Theorem l consists of two elements only). 3) We find statements in the literature which seem to contradict our last remarks. Thus, e.g., it is claimed in [5], pp. 343 f., that the betweenness relation B (or the order relation 0, which is closely related to B) can serve as the only primitive notion for three-dimensional Euclidean geometry, and this claim has been recently repeated in [6], pp. 8 and 38. However, when we analyze the argument in [5], pp. 381 ff., which is used to justify this claim, we notice that it actually leads to the opposite conclusion -to the statement that B cannot serve as the only primitive notion for Euclidean geometry. In fact, in [5], pp. 381 ff., it is proved correctly that, in terms of B, we can define various notions which have the same properties as the ordinary metric notions (such as 1), except that they are relativized to a few arbitrarily chosen points and vary with these points. Hence it is seen that Euclidean geometry (with any given number of dimensions) has various models in which the relation B is the same while the corresponding metric notions differ; by PADOA's method this implies at o~ce that the ordinary (non-relativized) metric notions cannot be defined in terms of B. This rather curious situation was noticed as far back as 1935; cf. [4], p. 88. 4) Compare the remarks conceming the axiom of choice in [1], in particular footnote 4.

474 BIBLI 0 G RAPHY I. BETH, E. W. and A. TxasKI, Equilaterality as the only primitive notion of

Euclidean geometry. Indagationes Mathematicae, 18, 462-467 (1956). 2. LINDENBAUM, A. and A. TxasKI, Uber die Beschriinktheit der Ausdrucksmittel deduktiver Theorien. Ergebnisse eines Mathematischen Kolloquiums, 7, 15-22 (1936). 3. Pmru, M., La geometria elementare istituita sulla nozioni di "punto" e "sfera". Memorie di Matematica e di Fisica della Societa Italiana della Scienze, ser. 3, 15, 345-450 (1908). 4. TxasKI, A., Einige methodologische Untersuchungen iiber die Definierbarkeit der Begriffe. Erkenntnis, 5, 80-100 (1935). · 5. VEBLEN, 0., A system of axioms for geometry. Transactions of the American Mathematical Society, 5, 343-384 (1904). 6. WILDER, R. L., Introduction to the foundations of mathematics (New York and London, 1952).

University of California, Berkeley, and John Simon Guggenheim Memorial Foundation.