Although there is no smallest ω model , Putnam and Gandy independently proved about 1963 that the class of ramified analytical sets, as defined by Cohen , form the smallest β model of analysis , . This paper will consider other restricted classes of models, namely the βn models for integers n > 1 , and prove under appropriate assumptions the existence of minimum such models. In fact we shall construct the minimum βn model in a fashion similar to the procedure yielding the class of ramified analytical sets, but adding at each stage a segment of the sets in those already obtained (for n a fixed integer > 1).
Roughly speaking a βn model is an ω model absolute for n-set-quantifier assertions about its subsets of natural numbers. A β model is simply a β1 model. See Enderton and Friedman  for a further investigation of βn models.