Handbook of Logic in Artificial Intelligence and Logic by Gabbay D.M., Hogger C.J., Robinson J.A. (eds.)

By Gabbay D.M., Hogger C.J., Robinson J.A. (eds.)

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Definition 2. A species that contains a denumerably infinite subspecies is called infinite. 40 SPREADS AND SPECIES Thus a species that cannot be finite, is not necessarily infinite. De fin it ion 3 . A species that is equivalent to a detachable subspecies of N is called numerable [L. E. J. Brouwer 1918, p. 7; 1924, p. 248], [A. Heyting 1929, p. 51]. Example. The species of twin primes (p, p+2) is numerable, though nobody knows whether it is finite or infinite. I shall not go into the theory of cardinal numbers, which differs much from the classical theory [L.

As to the notion of linear dependence, it can be defined in two ways; in (1) we can require the coefficients ,l to be # 0 or to be c:F 0; this gives respectively the notions of strong and of weak dependence. As the former is by far the most important, dependence without adjective will mean strong dependence. A system of vectors that cannot be dependent will be called independent. Just as in many other cases, besides this negative notion we can define a positive one, classically equivalent to it.

Cp2(a, b, c, ... ), ... be the functions for which successively the inverse must be taken in the calculation of f. Then cp1 (a, b, c, ... , ... , ... ) =1= 0 for n > k2 , etc. 4. def. l, that x,. , ... ) for n>k,. Theorem l. Every rational identity that is valid for rational numbers holds also in the following sense for real number-generators: Let f(p, q, r, .. , x, y, z, .. ) and g(p, q, r, .. , x, y, z, .. ) be rational functions such that f=g if for p, q, r, ... are substituted given rational numbers p 0 , %, r 0 , .

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