328 ANNUAL REPORT SMITHSONIAN INSTITUTION, 1961 



The important conclusion we can draw from the existence of a 

 model is that the system does not contain contradictions ; that is, it is 

 consistent. It is vital that a mathematical system be consistent ; for 

 if one can prove two contradictory statements in any logical system, 

 one can prove any statement whatever, and in an inconsistent system 

 any statement whatever is a theorem, so that the system is possibly 

 useless. 



From the point of view which obtained from the time of the Greeks 

 to the 19th century, the validity of Euclidean or non-Euclidean 

 geometry was thought to depend on their truth for points and lines 

 in physical space. From the point of view of modern mathematics, 

 this is a question of physics, not of mathematics ; it is a difficult one 

 for physics to settle, because even if non-Euclidean geometry is 

 physically valid its results for small enough figures differ little from 

 those of Euclidean geometry, just as it is impossible by studying a 

 small part of the earth to decide whether the earth is flat or round. 

 Actually the general theory of relativity teaches that non-Euclidean 

 geometry holds in the real world ; but this question in no way affects 

 the mathematical status of non-Euclidean geometry, though it cer- 

 tainly gives an added interest to its study. Even without this the 

 study of non-Euclidean geometry would not be an idle game; for 

 instance, Lobachevskian geometry has important applications in the 

 theory of functions of a complex variable. Validity of a mathemati- 

 cal system in the modern sense is a question of its consistency; if a 

 geometrical system is consistent, it is a worthwhile object of mathe- 

 matical study, and, experience has shown, it generally has application 

 in subjects where the "points" of the geometry may be entities far 

 removed in their nature from our intuitive idea of the points of space. 



Our proof that non-Euclidean geometry is consistent uses models 

 based on Euclidean geometry and therefore assumes implicitly that 

 Euclidean geometry is consistent. We can go on to construct a model 

 for Euclidean geometry in which the word "point" is interpreted to 

 mean an ordered pair (or, in the case of three-dimensional geometry, 

 an ordered trio) of real numbers ; this gives a proof of the consistency 

 of Euclidean geometry, but it assumes the consistency of the theory 

 of the real numbers. For the real numbers, again, models exist : the 

 most familiar is that in which a real number is taken to be an infinite 

 decimal, that is, a sequence of integers. Thus one can construct models 

 for the real numbers in terms of the integers and the theory of sets. 



Frege and Eussell and the school of mathematical logicians which 

 followed tliem aimed to go still further, and to eliminate the integers 

 as independent concepts by defining them in terms of sets. Thus one 

 can define "one" to be the set of all sets having only a single number, 

 "two" to be the set of all pairs, and so on ; the apparent circularity 



