441 research outputs found

    Algorithmic Randomness and Complexity

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    Things that can be made into themselves

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    One says that a property PP of sets of natural numbers can be made into itself iff there is a numbering Ξ±0,Ξ±1,…\alpha_0,\alpha_1,\ldots of all left-r.e. sets such that the index set {e:Ξ±e\{e: \alpha_e satisfies P}P\} has the property PP 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
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