The Recursion Theorem and Infinite Sequences

TL;DR

This paper employs the Recursion Theorem to demonstrate the existence of specific infinite sequences and sets with self-referential properties, advancing understanding of computably enumerable sets and their structures.

Contribution

It introduces new infinite sequences characterized by self-referential properties and establishes the concept of self-constructing sets, linking them to computably enumerable sets.

Findings

Existence of an increasing sequence where each set is the successor element.

Existence of an increasing sequence with sets containing all subsequent elements.

Every nonempty c.e. set disjoint from an infinite computable set is one-one equivalent to a self-constructing set.

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