The Recursion Theorem and Infinite Sequences
Logic
2008-01-15 v1
Abstract
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
Cite
@article{arxiv.0801.2097,
title = {The Recursion Theorem and Infinite Sequences},
author = {Arnold W. Miller},
journal= {arXiv preprint arXiv:0801.2097},
year = {2008}
}