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本文由Theodore A. Slaman撰写，总结了2005年新加坡数学科学研究所夏季学校数学逻辑课程中的讲座内容。文章探讨了图灵度D的全局结构、其自同构群的可数性以及图灵跳跃在D内的可定义性。特别关注了一个开放问题：图灵度是否存在非平凡的自同构？尽管Cooper在1999年声称找到了这样的自同构，但这一结果尚未得到独立验证。文中还讨论了Slaman-Woodin双解释猜想，即不存在这样的自同构。此外，文章还提供了关于图灵度自同构群Aut(D)的信息，证明了它是可数的，并且每个元素都是算术可定义的。这些结论为解决该领域未解问题提供了新的视角。

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Global Properties ofthe Turing Degrees and the
Turing Jump

Theodore A. Slaman∗

Department ofMathematics
University ofCalifornia, Berkeley
Berkeley, CA 94720-3840, USA
slaman@math.berkeley.edu

Abstract

We present a summary of the lectures delivered to the Institute for Mathematical Sci-

ences, Singapore, during the 2005 Summer School in Mathematical Logic. The lectures
covered topics on the global structure of the Turing degrees D, the countability of its
automorphism group, and the definability ofthe Turing jump within D.

1
Introduction

This note summarizes the tutorial delivered to the Institute for Mathematical Sciences,
Singapore, during the 2005 Summer School in Mathematical Logic on the structure ofthe
Turing degrees. The tutorial gave a survey on the global structure ofthe Turing degrees
D, the countability ofits automorphism group, and the definability ofthe Turing jump
within D.

There is a glaring open problem in this area: Is there a nontrivial automorphism of
the Turing degrees? Though such an automorphism was announced in Cooper (1999),
the construction given in that paper is yet to be independently verified. In this pa-
per, we regard the problem as not yet solved. The Slaman-Woodin Bi-interpretability
Conjecture 5.10, which still seems plausible, is that there is no such automorphism.

Interestingly, we can assemble a considerable amount of information about Aut(D),
the automorphism group ofD, without knowing whether it is trivial. For example, we
can prove that it is countable and every element is arithmetically definable. Further,
restrictions on Aut(D) lead us to interesting conclusions concerning definability in D.

Even so, the progress that can be made without settling the Bi-interpretability Con-
jecture only makes the fact that it is open more glaring. With these notes goes the hope
that they will spark further interest in this area and eventually a solution to the problems
that they leave open.

∗Slaman was partially supported by the National University ofSingapore and by National Science
Foundation Grant DMS-0501167. Slaman is also grateful to the Institute for Mathematical Sciences,
National University of Singapore, for sponsoring its 2005 program, Computational Prospects of
Infinity.

1

2. The coding lemma and the first order theory ofthe Turing degrees
2

1.1
Style

In the following text, we will state the results to be proven in logical order. We will
summarize the proofs when a few words can convey the reasoning behind them. When
that fails, we will try to make the theorem plausible. A complete discussion, including
proofs omitted here, can be obtained in the forthcoming paper Slaman and Woodin
(2005).

2
The coding lemma and the first order theory ofthe Turing degrees

Definition 2.1
• D denotes the partial order of the Turing degrees. a+ b denotes
the join oftwo degrees. (A⊕B denotes the recursive join oftwo sets.)
• A subset I ofD is an ideal ifand only ifI is closed under ≤T (x ∈I and y ≤T x
implies y ∈I) and closed under + (x ∈I and y ∈I implies x+ y ∈I). A jump
ideal is closed under the Turing jump (a 7→a0) as well.

Early work on D concentrated on its naturally order-theoretic properties. For ex-
ample, Klee...

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