Things that can be made into themselves

One says that a property P of sets of natural numbers can be made into itself iff there is a numbering α0, α1, . . . of all left-r.e. sets such that the index set {e : αe 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.