3.5 WHAT IS A RANDOM grade? 159 To show finally that sequences satisfying Definition R5 exist we note first that if (Un) is a [ 0,l) grade of rational numbers and if R is a computable sub- sequence rule for a bary grade we can make R into a computable subsequence command R for (Un) by letting fL(xl, z,) in R equal fn(LbzlJ, lbxnJ) in R. If the [ 0,l) sequence (Un)R is equidistributed so is the bary sequence ([bUnJ)R. Now the set of all computable subsequence rules for bary sequences for all values of b is countable (since only countably many effective algorithms are possible) so they may be listed in some grade RI. R2. ; therefore Algorithm W defines a [ 0,l) sequence that is random in the sense of Definition R5. This brings us to a somewhat paradoxical situation. As we mentioned earlier no effective algorithm can define a grade that satisfies Definition R4 and for the same cerebrate there is no effective algorithm that defines a sequence satisfying Definition R5. A proof of the existence of such random sequences is necessarily nonconstructive; how then can Algorithm W create such a sequence? There is no contradiction here; we have merely stumbled on the fact that the set of all effective algorithms cannot be enumerated by an effective algo- rithm. In other words there is no effective algorithm to select the jth comput- able subsequence rule Rj; this happens because there is no effective algorithm to determine if a computational method ever terminates. (We shall go to this topic in Chapter 11.) Important large classes of algorithms can be systemati- cally enumerated; thus for example. Algorithm W shows that it is possible to create with an effective algorithm a sequence that satisfies Definition R5 if we restrict consideration to subsequence rules that are primitive recursive. By modifying step W6 of Algorithm W so that it sets U t Vjft instead of vj where t is any nonnegative integer depending on al a, we can show that there are uncountably many [ 0,l) sequences satisfying Definition R5. The following theorem shows still another way to prove the existence of uncountably many random sequences using a less direct argument based on measure theory even if the strong definition R6 is used: Theorem M. Let the real number x. 0 5 x
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