那些可以自我改造的东西
Things that can be made into themselves
摘要：
有人说，自然数集合的性质$P$可以成为自身，当且仅当存在所有左r. e的编号$α_0,α_1,\ldots$。使得索引集$\{e: α_e$满足$P\}$的集合也具有属性$P$。例如，马丁-勒夫随机的性质可以变成它自己。在这里，我们描述了这些单例性质，可以成为自己。本工作的第二个方向是左-r. e结构的调查。集合在包含模有限集合下。与r.e.的相应结构相比，集合，其中只有极大但没有极小成员，极小和极大左-r. e。存在集合。此外，我们的最小和最大左-r.e.集合与Friedberg的最大r.e.的经典构造有很大不同。集.最后，我们研究了极小和极大左r.e.集可以自己制作。
英文摘要：
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.
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