Things that can be made into themselves

Abstract

One says that a property $P$ of sets of natural numbers can be made into itself iff there is a numbering $α_0,α_1,\ldots$ of all left-r.e. sets such that the index set $\{e: α_e$ satisfies $P\}$ has the property $P$ as well. For example, the property of being Martin-Löf 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.