PureMath.pdf

Be one of those who prefer to omit the discussion of

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be one of those who prefer to omit the discussion of the notion of an irrational number contained in §§ 6 12 .

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[I : 17] REAL VARIABLES 31 may not suffice to define one of the real numbers. This is so, for example, with the pair ‘ x < 2’ and ‘ x > 2’ or (if we confine ourselves to positive numbers) with ‘ x 2 < 2’ and ‘ x 2 > 2’. Every rational number possesses one or other of the properties, but not every real number, since in either case 2 escapes classification. There are now two possibilities. * Either L has a greatest member l , or R has a least member r . Both of these events cannot occur. For if L had a greatest member l , and R a least member r , the number 1 2 ( l + r ) would be greater than all members of L and less than all members of R , and so could not belong to either class. On the other hand one event must occur. For let L 1 and R 1 denote the classes formed from L and R by taking only the rational members of L and R . Then the classes L 1 and R 1 form a section of the rational numbers. There are now two cases to distinguish. It may happen that L 1 has a greatest member α . In this case α must be also the greatest member of L . For if not, we could find a greater, say β . There are rational numbers lying between α and β , and these, being less than β , belong to L , and therefore to L 1 ; and this is plainly a contradiction. Hence α is the greatest member of L . On the other hand it may happen that L 1 has no greatest member. In this case the section of the rational numbers formed by L 1 and R 1 is a real number α . This number α must belong to L or to R . If it belongs to L we can show, precisely as before, that it is the greatest member of L , and similarly, if it belongs to R , it is the least member of R . Thus in any case either L has a greatest member or R a least. Any section of the real numbers therefore ‘corresponds’ to a real number in the sense in which a section of the rational numbers sometimes, but not always, corresponds to a rational number. This conclusion is of very great importance; for it shows that the consideration of sections of all the real numbers does not lead to any further generalisation of our idea of number. Starting from the rational numbers, we found that the idea of a section of the rational numbers led us to a new conception of a number, that of a real number, more general than that of a rational number; and it might have * There were three in § 6 . This was not the case in § 6 .
[I : 18] REAL VARIABLES 32 been expected that the idea of a section of the real numbers would have led us to a conception more general still. The discussion which precedes shows that this is not the case, and that the aggregate of real numbers, or the continuum, has a kind of completeness which the aggregate of the rational numbers lacked, a completeness which is expressed in technical language by saying that the continuum is closed.

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• Fall '14

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