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 , . . . 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.Here the additive log factor is used for coding two implicit programs into a single string. On the other hand, by the definition of f , H(A ↾↾ m) > m + 3r(m)for all m, a contradiction. Therefore I is indifferent for A.

Arithmetic complexity via effective names for random sequences

“…We summarize our main results in Table 1. A theorem in a forthcoming paper [28] states that every ∅ ′ -recursive 1-generic set has a co-r.e. indifferent set which is retraceable by a recursive function.…”

“…real cannot be 1-generic [22,Proposition XI.2.3], weakly 1generic sets can be left-r.e. [10,20,28]. We shall make use of the following result of Kurtz which also appears in [7,Theorem 8.11.7].…”

“…We summarize our main results in Table 1. A theorem in a forthcoming paper [28] states that every ∅ ′ -recursive 1-generic set has a co-r. 1 simply by observing differences in arithmetic complexity (and using the well-known result Theorem 1.3). The remaining separations follow from Theorem 5.1 and Proposition 4.5.…”