可数图灵理想的恰对和一致上界

一个（图灵）理想，是满足两个封闭条件的图灵度集合：向下封闭；任意，中一对图灵度的上确界也在，中。可数理想不仅在图灵度整体性质的研究中有着重要意义，而且在对哥德尔可构成集合L精细结构的早期研究中也发挥过重要作用。研究可数理想的两个重要概念是：恰对和一致上界。借助这两个概念，我们可以将可数理想简化为一个（一致上界）或者一对（恰对）图灵度。通过前人的研究，我们可以发现这两个概念是紧密相连的，同时我们也可以对它们的关系提出进一步的问题。在本文中，我们证明以下定理：任给一个可数理想I，都存在两个I的一致上界a0和a1，同时a0和a1构成，的一个恰对。此定理从正面回答了Lerman提出的关于算术图灵度构成的理想的一个问题。此定理的证明实际上是经过小心修改的、典型的恰对构造。我们在典型恰对构造的过程中，加入一些微妙的限制，使得形成恰对的两个图灵度a0和a1可以各自独立地在一定程度上用逼近的办法还原整个构造，从而分别给出可数理想I的一致枚举。在a0和a1分别的逼近中，我们引入了有穷损坏方法。本文的最后指出a0和a1的图灵跃迁的一些性质。

Exact Pairs and Uniform Upper Bounds

A （Turing） ideal I is a downward closed set of Turing degrees which is also closed under the supremum operation. Countable ideals have played interesting roles in the studies of global aspects of the partial order of Turing degrees, and also in early development of fine structure theory of Godel＇s constructible universe. Uniform upper bounds and exact pairs are two useful notions in studying countable ideals. They reduce countable ideals which are sets of Turing degrees to finite tuples of Turing degrees. Previous works have shown that these two notions are very closed and lead to further questions on interactions between them. In this paper, we prove that for any countable ideal 1 of degrees, there are a0 and a~ such that they are uniform upper bounds of functions of degrees in I and form an exact pair of I. This result affirmatively answers a question raised by Lerman on I being the set of arithmetic Turing degrees. The proof is actually a deliberate adaptation of the typical exact pair construction. Subtle settings make components of an exact pair capable to solely approximate the construc- tion to some extent, and thus also capable to uniformly enumerate representatives of elements in a given ideal. Finite injury argument is to be used for the approximation. Finally, we remark on the jumps of uniform upper bounds constructed in this paper.