In this paper we use the Recursion Theorem to show the existence of various infinite sequences and sets. Our main result is that there is an increasing sequence e_0, e_1, e_2 .. such that W_{ e_n}={ e_{ n+1}} for every n. Similarly, we prove that there exists an increasing sequence such that W_{ e_n}={ e_{ n+1},e_{ n+2},...} for every n. We call a nonempty computably enumerable set A self-constructing if W_e=A for every e in A. We show that every nonempty computable enumerable set which is disjoint from an infinite computable set is one-one equivalent to a self-constructing set
Categories: math.LO