Computably Enumerable Reals and Uniformly Presentable Ideals

We study the relationship between a computably enumerable real and its presentations. A set A presents a computably enumerable real α if A is a computably enumerable prefix‐free set of strings such that $ \alpha = \sum _{\sigma \in A} 2 ^{- \vert \sigma \vert} $ . Note that $ \alpha = \sum _{\sigma \in A} 2 ^{- \vert \sigma \vert} $ is precisely the measure of the set of reals that have a string in A as an initial segment. So we will simply abbreviate $ \alpha = \sum _{\sigma \in A} 2 ^{- \vert \sigma \vert} $ by μ ( A ). It is known that whenever A so presents α then A ≤ wtt α , where ≤ wtt denotes weak truth table reducibility, and that the wtt‐degrees of presentations form an ideal ℐ( α ) in the computably enumerable wtt‐degrees. We prove that any such ideal is $ \sum ^{0} _{3} $ , and conversely that if ℐ is any nonempty $ \sum ^{0} _{3} $ ideal in the computably enumerable wtt‐degrees then there is a computable enumerable real α such that ℐ = ℐ( α ). We also prove a kind of Rice Theorem for these ideals, namely that if the index set of such a $ \sum ^{0} _{3} $ ideal is not empty or equal to ω then it is $ \sum ^{0} _{3} $ ‐complete.