Computing on the Banach space C [ 0 , 1 ]

We demonstrate that, within any computable presentation of the Banach space C [ 0 , 1 ], computing 1 is no harder than computing the halting set. Additionally, we prove that the modulus operator | · | is Ø ″ -computable and use this to show that C [ 0 , 1 ] is Δ 3 0 -categorical when we restrict ourselves to the presentations in which at least one homeomorphism of the unit interval onto itself is computable.

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.