We give a straightforward computable-model-theoretic definition of a property of \Delta^0_2 sets called order-computability. We then prove various results about these sets which suggest that, simple though the definition is, the property defies any easy characterization in pure computability theory. The most striking example is the construction of two computably isomorphic c.e. sets, one of which is order-computable and the other not.
"Order-Computable Sets." Notre Dame J. Formal Logic 48 (3) 317 - 347, 2007. https://doi.org/10.1305/ndjfl/1187031407