Axiomatizability of Geometry without points. by Andrzej GrzegorczykReview by: Wolfram SchwabhäuserThe Journal of Symbolic Logic, Vol. 37, No. 1 (Mar., 1972), p. 201Published by: Association for Symbolic LogicStable URL: http://www.jstor.org/stable/2272626 .

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REVIEWS 201

i = m) without leaving S, after a finite number of moves ? 1. To state such conditions for the general case, however, is much more complicated. The proof is therefore long, but quite clearly written.

The paper also contains an interesting discussion of relevant aspects of infinite game theory and operators that constitute strategies for such games. Although the authors mention the reviewer's name favorably in two connections, they kindly omitted reference to the reviewer's earlier fallacious proof of their theorem, which was unpublished but unfortunately distributed in dittoed form. Landweber was the one who detected the subtle error in that proof.

A slight theoretical oversight occurs at the bottom of page 307. The possibility must be provided for in (f) that n > 0 but Zt' f Al, in which case Vt' should be put equal to [ ].

Errata: Page 305, line 10, R1 should be RI . Page 305, line 28, hl should be hi. Page 306, line 24, Vf should be Vt. Page 307, line 26, U should be set union. Page 308, line 13, "from I- sequences to J-sequences" should be "from J-sequences to I-sequences."

ROBERT MCNAUOHTON

ANDRzEj GRZEGORCZYK. Axiomatizability of geometry without points. Synthese, vol. 12 nos. 2-3 (1960), pp. 228-235; also in The concept and role of the model in mathematics and natural and social sciences, Synthese Library, D. Reidel Publishing Company, Dordrecht 1961, pp. 104-111.

The author introduces a geometry T without points by giving a finite (second-order) axiom system having the primitive notions P1(x) ("x is a spatial body"), x c y (" the body x is con- tained in the body y"), and x I y ("the bodies x, y are separated"). A theory S of topology with primitive notions Qj(p) ("p is a point") and Q2(x) ("x is an open set of points") is based on axioms saying that (Qi, Q2) is a Hausdorff space and each point can be obtained as intersection of a family of open domains of a certain kind.

An interpretation of Tin S is obtained by defining spatial bodies as non-empty open domains, c as set-theoretical inclusion- of bodies, and x I y as disjointness of the closures of x and y. On the other hand, a point can be characterized by the set of all spatial bodies containing it, and it turns out that such sets of bodies can be described in T already. Thus, also an interpretation of S in Tis obtained by defining (pseudo-)points as such sets of bodies and defining open sets in a natural way. Starting from T and applying both interpretations, one gets an isomorphism between the original bodies and the new "' pseudo-bodies " (non-empty open domains of pseudo- points).

It is concluded that the set S of topological consequences of T (formulated with predicates QG, Q2 only) (i) is contained in S, and (ii) is finitely axiomatizable (with axioms obtained from those of T). Some general theorems about mutual interpretability of theories are provided for this at the beginning. The reviewer suggests that (ii) can also be obtained by observing that 3 equals S, which is already finitely axiomatized. WOLFRAM SCHWABHXUSER

WANDA SzmIELwW. New foundations of absolute geometry. Logic, methodology and philos- ophy of science, Proceedings of the 1960 International Congress, edited by Ernest Nagel, Patrick Suppes, and Alfred Tarski, Stanford University Press, Stanford, Calif., 1962, pp. 168-175.

The author extends her absolute calculus of segments construction, which was developed in XXVII 238, to the theory -V of (2-dimensional) absolute geometry without the continuity axioms. Recall that an ordered field is Euclidean if every positive element is a square. In the notation of the review XXVII 238, the system <S, +, *, S> can be embedded in a commutative Euclidean field as an open interval which contains the unit element Uand has the field zero as an endpoint. This embedding is essentially unique, and provides a representation theorem for models of d. The construction is given in slightly less detail than in the earlier article, and the representation theorem is not explicitly stated.

Misprints: Page 170, line 2, replace "P(XYX1 Y)" by "P(XYX1 Y1)"; page 172, third line from the bottom, in the phrase " since + is not always performable" circle the " + ".

THOMAS FRAYNE

JON BARWISE. Implicit definability and compactness in infinitary languages. The syntax and semantics of infinitary languages, edited by Jon Barwise, Lecture notes in mathematics, no. 72, Springer-Verlag, Berlin, Heidelberg, and New York, 1968, pp. 1-35.

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