The goal of this paper is to announce there is a single orbit of the c.e. sets with inclusion, $\E$, such that the question of membership in this orbit is $Σ^1_1$-complete. This result and proof have a number of nice corollaries: the Scott rank of $\E$ is $\wock +1$; not all orbits are elementarily definable; there is no arithmetic description of all orbits of $\E$; for all finite $α\geq 9$, there is a properly $Δ^0_α$ orbit (from the proof).
A few small corrections made in this version
本文的目的是要证明有一个单一的轨道的c.e.包含的集合，$\mathcal{E}$，使得这个轨道中的成员资格问题是$Σ^1_1$ -完全的。这个结果和证明有许多很好的推论：$\mathcal{E}$的Scott秩是$ω^{CK}_1+1; Not all orbits are elementarily definable; There is no arithmetic description of all orbits of $ \mathcal{E} $; For all finite $ α \geq 9 $, there is a properly $ Δ^0_α$轨道（来自证明）。2007年4月6日，微小变更2007年11月20日，微小变更
We investigate what collections of c.e.\ Turing degrees can be realised as the collection of elements of a separating $Π^0_1$ class of c.e.\ degree. We show that for every c.e.\ degree $\mathbf{c}$, the collection $\{\mathbf{c}, \mathbf{0}'\}$ can be thus realized. We also rule out several attempts at constructing separating classes realizing a unique c.e.\ degree. For example, we show that there is no \emph{super-maximal} pair: disjoint c.e.\ sets $A$ and $B$ whose separating class is infinite, but every separator of c.e.\ degree is a finite variant of either $A$ or $\overline{B}$.
A note on the differences of computably enumerable reals
我们表明，给定任何不可计算的左c. e。实$α$存在一个左c. e.实际$β$，使得对于所有左c. e. reals and all right-c.e.真正的$γ$。证明是不一致的，二分法是给定的真实$α$是否是马丁-勒夫随机或没有。由此可见，给定任何通用机$U$，存在另一个通用机$V$，使得U的停机概率不是V的停机概率的左c. e的平移。真的我们不知道这一事实是否有统一的证据。
The study of automorphisms of computable and other structures connects computability theory with classical group theory. Among the noncomputable countable structures, computably enumerable structures are one of the most important objects of investigation in computable model theory. In this paper, we focus on the lattice structure of computably enumerable substructures of a given canonical computable structure. In particular, for a Turing degree $\mathbf{d}$, we investigate the groups of $\mathbf{d}$ -computable automorphisms of the lattice of $\mathbf{d}$-computably enumerable vector spaces, o…
Institute of Discrete Mathematics and Geometry, TU Wien（维也纳工业大学离散数学与几何研究所）；Department of Humanistic Research and Innovation, University of Bari（巴里大学人文研究与创新系）
