In the first half of this paper, we study the way that sets of real numbers closed under Turing equivalence sit inside the real line from the perspective of algebra, measure and orders. Afterwards, we combine the results from our study of these sets as orders with a classical construction from Avraham to obtain a restriction about how non trivial automorphism of the Turing degrees (if they exist) interact with 1-generic degrees.

中文翻译：

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图灵等价下封闭的实数集：域、阶和自同构的应用

在本文的前半部分，我们从代数、测度和阶的角度研究了图灵等价下实数集闭合的方式。之后，我们将这些集合作为阶的研究结果与 Avraham 的经典构造相结合，以获得关于图灵度（如果存在）的非平凡自同构如何与 1-泛型度相互作用的限制。