Undecidability of Some Topological Theories by Andrzej GrzegorczykReview by: Hartley Rogers, Jr.The Journal of Symbolic Logic, Vol. 18, No. 1 (Mar., 1953), pp. 73-74Published by: Association for Symbolic LogicStable URL: http://www.jstor.org/stable/2266343 .

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REVIEWS 73

the question whether only provable formulas have this property. In fact he raised the question whether there does not exist a particular model (IOs, BO) with respect to which each non-provable formula takes on a non-zero value. The corresponding question for classical calculus is known to have an affirmative answer from the work of G6del (4182); Dr. Rasiowa has now shown that the affirmative answer is likewise true for the intuitionistic calculus.

A partial answer in this direction was previously reported on by Henkin (XVI 290). Giving a unified treatment which could be applied to the classical, intuitionistic, and modal cases, Henkin's result fell short of answering Mostowski's question only in that the algebra Bo of his model is not complete- although sums and products exist for all subsets of Bo needed in carrying through Mostowski's interpretation. In the present paper Rasiowa goes through the same construction (however, this part of her work was done independently of Henkin's), obtaining a denumerable but incomplete Brouwerian lattice as model for the Heyting calculus. She then goes on to show that every Brouwerian lattice can be extended to a complete Brouwerian lattice preserving all (finite and infinite) sums and products, thus finally answering Mostowski's question in every detail. The last step is accomplished as follows. The given Brouwerian lattice B is represented as the lattice of closed elements in a closure algebra C, using a result of McKinsey and Tarski (XI 83). Then C, considered as a Boolean algebra, is extended to a complete algebra C1, preserving all sums and products of C, by use of MacNeille's methods (III 58); after which the closure operation of C is extended to C1 by a method originally due to McKinsey (VII 118). Finally, the Brouwerian lattice of closed elements of C1 is shown to be the required completion of the given lattice B.

The author also shows that a modal functional calculus based on the system S4 of Lewis and Langford (4561) is complete (in the same sense as above) when interpreted with respect to models (I, C) in which C is a closure algebra. The relationship between the modal calculus considered by Rasiowa and those discussed earlier by Barcan (XI 96) and Carnap (XIII 218) is not made clear. LEON HENKIN

ANDRZEJ GRZEGORCZYK. Undecidability of some topological theories. Ibid., pp. 137 -152.

In an abstract in this JOURNAL, VOL. 14 (1949), pp. 75-76, Tarski presented the ingenious and useful notion of essential undecidability. An elementary theory is said to be essentially undecidable if it has no consistent extension (by addition of axioms or primitive relations or both) which is decidable. Tarski and Mostowski exhibit an essentially undecidable theory To which is also finitely axiomatizable. To can then be used to show undecidability of more general theories, for (as is easily shown via deduction theorem) any elementary theory T1 in which To is consistently interpretable or which has a common consistent extension with To is undecidable. A particularly useful feature of this technique is that once To has been shown interpretable in T1, then axiomatic assumptions (but not primitive relations) can be deleted from T1 while interpretability and hence undecidability remain.

In the present paper, Grzegorczyk makes a straightforward application of this method to demonstrate the undecidability of the elementary theories of certain top- ological algebras; He begins by presenting an arithmetic of finite sets, T, in which the individuals are finite point sets, and the relations are, in effect, relations of cardinality with appropriate equivalences introduced. This theory is seen to be isomorphic to the finitely axiomatizable, essentially undecidable arithmetic of R. M. Robinson (see Proceedings of the International Congress of Mathem-aticians 1950, vol. 1, pp. 729-730) and is hence itself finitely axiomatizable and essentially undecidable.

This means that any algebra of point sets, S, in which finite sets and arithinetic operations upon them can be defined (and hence in which T can be consistently

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74 REVIEWS

interpreted) is undecidable. Furthermore we may delete any assumptions from such an S. provided we keep all primitive relations, and the theory will remain undecidable. Thus in showing the undecidability of the general closure algebra C, the author adds assumptions to obtain a closure algebra S of a rather special sort (separable, normal, connected, etc.) in which T can be defined in terms of the closure primitives, and this suffices to show the undecidability of C. The undecidability of C is a result of Ja6- kowski. The result here is in fact the stronger one that any closure algebra consistent with S is undecidable. Thus the closure algebra of Euclidean n-space, n 2 2, proves to be undecidable and hence, since this algebra is complete with respect to truth, unaxiomatizable.

By the same method the author shows the undecidability of the elementary theories of certain other algebras: (1) the Brouwerian algebra of closed point sets (a Boolean algebra except that the difference between two sets is taken to be the closure of the Boolean difference); (2) the algebra of bodies (Boolean algebra plus a primitive to be interpreted as 'mutually tangent'); (3) the algebra of convexity (Boolean algebra in a Euclidean space together with a primitive for 'convex closure'); (4) semi-projective algebras (Boolean algebras together with n operators C,, ... , C. having axiomatic properties similar to those of semi-projection in Euclidean n-space-where semi- projection means that Ci of a point is the half-line projected from that point in negative direction parallel the ith axis).

A number of typographical errors and oversights have occurred. Some of the less obvious are as follows: (1) p. 137, lines 20-21, the definition of logical type is to be emended to 'a type of order n+1 is a subclass of the types of order n'; (2) p. 141, lines 10 and 19, change 'last' to 'next to last'; (3) p. 144, footnote, change '150' to '130'; (4) p. 146, line 3 change 'D -A-' to '-a- D'; and (5) p. 151, line 22 change 'CDC2C' to 'C2DCC'; line 23 change 'CHC2I' to 'C2HCI'. In the section on Brouwerian algebra, the writer asserts that every such algebra is a projective geometry. This is erroneous since the latter must be a complemented lattice, while the former need not be. Thus the undecidability of projective geometry algebra (shown by Tarski) does not follow as a corollary to the author's work.

Grzegorczyk's techniques are clear and possess evident power. Certain open problems are mentioned which the reader may find of interest: (1) decidability of closure algebra in Euclidean 1-space; (2) decidability in semi-projective convex algebras; (3)-(6) de- cidability of topological algebras based on the class of boundary sets; of dense sets; of non-dense sets; or of dense-in-themselves sets as primitive. The reviewer would like to point out that for case (3) the theory is undecidable since, in Euclidean 2-space, the class of closed sets is the same as the class of boundary sets, and the axioms of Euclidean 2-space are consistent with those of the special closure space constructed by Grze- gorczyk. HARTLEY ROGERS, Jr.

HEINRICH BEHMANN. Das Aufhlsungsproblemn in der Klassenlogik. Archiv fur mathlematische Logik und Grundlagenforschung, vol. 1 no. 1 (1950), pp. 17-29, and vol. 1 no. 2 (1951), pp. 33-51; also Archiv fur Philosophie, vol. 4 no. 1 (1950), pp. 97-109, and vol. 4 no. 2 (1951), pp. 193-21 1.

The Auflosungsproblemn in the logic of classes, in the case of one unknown, may be stated as follows: given a formula F(al . an, o) with the class variables a , an, Q as the only free variables occurring in it, to find a formula - =/(a,

an, -r) such that, whenever the condition (3L)F(al, a, a ) is fulfilled, it must yield, given a system of classes as values for al, .. an , all and only those values of Q which satisfy F(al, . . ., an, Q). It is supposed of course that /(a1, ;. ., an , x) is a designatory expression built from al, . . ., an, x by means of the symbolism of the logic of classes. This mode of considering the problem was adopted by Schr6der

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