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 apsects of the partial order of Turing degrees, and also in early development of fine structure theory of Gödel’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 I of degrees, there are a0 and a1 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 construction to some extent, and thus also capable to uniformly enumerate reprentatives 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.