Title: The Recursion Theorem and Infinite Sequences
Author: Arnold W. Miller
Abstract:
In this paper we use the Recursion Theorem to show the
existence of various infinite sequences and sets. Our
first 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
Latex2e: 7 pages
Latest version at: www.math.wisc.edu/~miller