16 ERRETT A. BISHOP Two real numbers {a } and {b 3 (the corresponding convergence n n parameters (b) and proofs (c) are assumed without explicit mention) are said to be equal if for each positive integer k there exists a positive integer N 1 k such that I an - bnl s - whenever n 2 N It can be shown that this notion k k ' of equality is an equivalence relation. Addition and multiplication of real numbers are also defined in the same way as they are defined classically. The order relation, on the other hand, is more interesting. If a = Isn} and b = {bn 1 are real numbers. we define a C b to mean that there exist posi- tive integers M and N such that a s bn - - whenever n 2 N. Then it is n M easily shown that a C b and b c imply a c, that a b implies a - c C b - c, and so forth. Some care must be exercised in defining the rela- tion 5 . We could define a 5 b to mean that either a b or a = b. An alternate definition would be to define it to mean that b C a is contradictory. We shall not use either of these, although our definition turns out to be equiva- lent to the latter. DEFINITION. a b means that for each positive integer M there exists a positive integer N such that b 2 a - - whenever n 2 N. n n M To make the choice of this definition plausible, I shall construct a cer- tain real number H . where an = 0 in case every even integer between 4 and n is the sum of two primes, and an = 1 otherwise. (More precisely, H is given by the Cauchy sequence {an} , with and the sequence [N 1 of convergence parameters, where N = n. ) Clearly n n we wish to have H 2 0. It certainly is according to the definition we have chosen. (The real number 0 of course is the Cauchy sequence of rational numbers all of whose terms are 0. ) On the other hand, we would not be en- titled to assert that H 2 0 if we had defined H 2 0 to mean that either H 0 or H = 0, because the assertion "H 0 or H = 0" means that we have a finite, purely routine method for deciding which in this case, a finite, purely
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