We examine model-theoretic properties of U-Prod N where U is a non-principal ultrafilter on w, and N is the structure w together with all its finitary functions and relations. This structure is w(,1)-saturated, hence saturated if CH holds. We examine what can occur in models of ZFC + ('(NOT))CH. The first model we consider is the one obtained by adding k (GREATERTHEQ) w(,2) Cohen reals to a model of ZFC + GCH. Here, we show that for all regular uncountable cardinals a,b (LESSTHEQ) k there exist ultrafilters U(,a,b) such that the non-st and ard part of U-Prod(w, w(,w), there will be continuum many of these. Moreover when k (GREATERTHEQ) w(,3) we can get continuum many ultrafilters whose ultraproducts of (w,+,*) are non-isomorphic. (Under CH, all such ultraproducts are isomorphic). We also consider the model obtained by adding k (GREATERTHEQ) w(,2) r and om reals to a model of ZFC + GCH. It is easy to see that all ultraproducts in this model will have cofinality = w(,1), which precludes saturation. We prove the existence of ultraproducts with the following saturation property: they consist of a saturated model of the theory of N plus a top sky. The properties of these ultrafilters are discussed. This seems to be the maximal amount of saturation possible in this model: There are Dedekind cuts in the top sky of every ultraproduct where the cofinality of the lower segment and the coinitiality of the upper segment are both w(,1). Also we show that in this model there are, for any uncountable regular a, ultrafilters U such that a is the coinitiality of the non-st and ard part of U-Prod(w,<).Ph.D.MathematicsUniversity of Michiganhttp://deepblue.lib.umich.edu/bitstream/2027.42/158962/1/8224919.pd
