Degrees of enumerations of countable Wehner-like families

The paper is a survey of results on countable families with natural degree spectra. These results were obtained by a modification of the methodology proposed by Wechner, who first found a family of sets with the spectrum consisting precisely of nonzero Turing degrees. Based on this method, many researchers obtained examples of families with another natural spectra. In addition, the paper extends these results presenting new examples of natural spectra. In particular, a family of finite sets with the spectrum consisting of exactly non-$K$-trivial degrees are constructed, and also we find new sufficient conditions on $\Delta^0_2$-degree $\mathbf{a}$ which guarantees that the class $\{\mathbf{x}: \mathbf{x}\not\leqslant\mathbf{a}\}$ is the degree spectrum of some family. Finally, we give a survey of our recent results on the degree spectra of $\alpha$-families, where $\alpha$ is an arbitrary computable ordinal.