Consider the ordering \(\omega\) of the natural numbers \(0\leq 1\leq 2\leq 3\leq\dots\). We can build an isomorphic order \(\mathcal B\) (for bad) as follows: We order all the even numbers in ascending order and enumerate the Halting set. If \(n\) enters the Halting set at some stage \(s\), then we put an odd number greater than \(s\) into \(\mathcal B\) between \(2n\) and \(2n+2\). You can convince yourself, that the order \(\mathcal B\) is computable and that every isomorphism between \(\mathcal B\) and \(\mathcal N\) will compute the Halting set. Why is that?
Well, in order to compute isomorphisms between copies of \(\omega\) we need to know the successor relation. For \(\omega\) this relation is computable while for \(\mathcal B\), it computes the Halting set. It turns out that \(\mathbf 0'\), the Turing degree of the Halting set, is the least degree computing isomorphims between any two computable copies of \(\omega\). We say that \(\omega\) has degree of categoricity \(\mathbf 0'\). Degrees of categoricity are a robust measure of the algorithmic complexity of a computable structure: Up to \(\mathbf 0'\) all the copies of \(\omega\) look the same and it can compute isomorphisms between them.
For over ten years degrees of categoricity have been heavily studied by computability theorists. The main questions are:
- Which Turing degrees are degrees of categoricity of computable structures?
- Say d is a degree of categoricity, can we always build two isomorphic computable structures A and B with degree of categoricity d and such that d is the least degree computing isomorphism between A and B? Such degrees are called strong.
In a joint project with Barbara Csima we made a breakthrough on these two questions by characterizing the degrees of categoricity above \(\mathbf 0''\) and by showing that every degree of categoricity above \(\mathbf 0''\) is strong. We do this by showing that every degree of categoricity is treeable, that is, the degree of the Turing least path through a computable tree, and that the treeable degrees above \(\mathbf 0''\) are all degrees of categoricity. Here is the preprint: http://arxiv.org/abs/2209.04524