Things that can be made into themselves

TL;DR

This paper characterizes singleton properties of sets of natural numbers that can be made into themselves through specific numberings, and explores the structure of left-r.e. sets, including minimal and maximal elements, and their self-referential properties.

Contribution

It provides a characterization of singleton properties that can be made into themselves and analyzes the structure of left-r.e. sets, including the existence of minimal and maximal sets, contrasting with classical r.e. sets.

Findings

Characterization of singleton properties that can be made into themselves.

Existence of both minimal and maximal left-r.e. sets under inclusion modulo finite sets.

Construction methods for minimal and maximal left-r.e. sets differ from classical r.e. set constructions.

Abstract

One says that a property $P$ of sets of natural numbers can be made into itself iff there is a numbering $\alpha_0,\alpha_1,\ldots$ of all left-r.e. sets such that the index set $\{e: \alpha_e$ satisfies $P\}$ has the property $P$ as well. For example, the property of being Martin-L\"of random can be made into itself. Herein we characterize those singleton properties which can be made into themselves. A second direction of the present work is the investigation of the structure of left-r.e. sets under inclusion modulo a finite set. In contrast to the corresponding structure for r.e. sets, which has only maximal but no minimal members, both minimal and maximal left-r.e. sets exist. Moreover, our construction of minimal and maximal left-r.e. sets greatly differs from Friedberg's classical construction of maximal r.e. sets. Finally, we investigate whether the properties of minimal and maximal left-r.e. sets can be made into themselves.