I'm reading this paper: http://www.math.cornell.edu/~shore/papers/pdf/hyp9.pdf and I am struggling with the meaning of Biintereptability, to quote the paper
A degree structure $D$ is biinterpretable with second order arithmetic if there is a definable standard model of arithmetic (or class of structures all isomorphic to $\mathbb{N}$) with definable schemes for both quantification over subsets of the model and a relation matching degrees with codes for sets in the model which are of the specified degrees.
I feel that the intuative meaning should be that there is a way of interpreting 2nd order arithmetic in the degree structure $D$ in some relatively effective way and the same with $D$ and arithmetic reversed. However I really don't get what the formal definition is saying.
Any illuminating thoughts?