Reunion

Over certain transitive models of Z, the usual treatment of forcing goes awry. But the provident closure of any such set is a provident model of Z, over which, as shown in "Provident sets and rudimentary set forcing", forcing works well. In "The Strength of Mac Lane Set Theory" a process is described of passing from a transitive model of Z + Tco to what is here called its lune, which is a larger model of Z + KP.

Theorem: Over a provident model of Z, the two operations of forming lunes and generic extensions commute.

Corresponding results hold for transitive models of Mac Lane set theory + Tco.