Selection by Recursively Enumerable Sets

For given sets A, B and Z of natural numbers where the members of Z are z 0, z 1, … in ascending order, one says that A is selected from B by Z if A(i) = B(z i ) for all i. Furthermore, say that A is selected from B if A is selected from B by some recursively enumerable set, and that A is selected from B in n steps iff there are sets E 0,E 1,…,E n such that E 0 = A, E n = B, and E i is selected from E i + 1 for each i < n.

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