Degrees of and lowness for isometric isomorphism

We contribute to the program of extending computable structure theory to the realm of metric structures by investigating lowness for isometric isomorphism of metric structures. We show that lowness for isomorphism coincides with lowness for isometric isomorphism and with lowness for isometry of metric spaces. We also examine certain restricted notions of lowness for isometric isomorphism with respect to fixed computable presentations, and, in this vein, we obtain classifications of the degrees that are low for isometric isomorphism with respect to the standard copies of certain Lebesgue spaces.

m-topped degrees

This chapter assesses m-topped degrees. The notion of m-topped degrees comes from a general study of the interaction between Turing reducibility and stronger reducibilities among c.e. sets. For example, this study includes the contiguous degrees. A c.e. Turing degree d is m-topped if it contains a greatest degree among the many one degrees of c.e. sets in d. Such degrees were constructed in Downey and Jockusch. The dynamics of the cascading phenomenon occurring in the construction of m-topped degrees strongly resemble the dynamics of the embedding of the 1–3–1 lattice in the c.e. degrees. Similar dynamics occurred in the original construction of a noncomputable left–c.e. real with only computable presentations, which was discussed in the previous chapter.

The computable dimension of trees of infinite height

We prove that no computable tree of infinite height is computably categorical, and indeed that all such trees have computable dimension ω. Moreover, this dimension is effectively ω, in the sense that given any effective listing of computable presentations of the same tree, we can effectively find another computable presentation of it which is not computably isomorphic to any of the presentations on the list.