Methods for carrying out transfinitely nested priority constructions have been developed by Harrington  and by Ash [2, 1, 3, 4]. Ash's method has different versions, with later ones becoming simpler. Lemmp and Lerman  have also developed a method, for finitely nested constructions. Ash formulated abstractly the object of a nested priority construction, and he proved a metatheorem for what he called “α-systems”, listing conditions which guarantee the success of the construction. Harrington's method of “workers”, at least in its original, informal state, seems more flexible than Ash's α-systems.
In [10, 9], there are finite and transfinite versions of a metatheorem for workers. The statements are complicated, and these metatheorems have not proved to be very useful. The present paper gives a new transfinite metatheorem. The statement is considerably simpler than the one in , although not so simple as that in . The new metatheorem grew, in part, out of an effort to find a new proof of Ash's metatheorem. The new metatheorem yields the one in , and it seems more flexible. A different generalization of Ash's metatheorem will be given in .
Ash's metatheorem is easier to use than the one in the present paper, and the result in  is certainly the one to use wherever it applies. Here we give one application of the new metatheorem which does not seem to follow from the result in . This is a theorem on models “representing” a given Scott set, which implies one half of a recent result of Solovay , on Turing degrees of models of particular completions of Peano arithmetic (PA).