Appendix to “Iterated Ultrapowers for the Masses”

This document provides solutions to the exercises in “Iterated Ultrapowers for the Masses”. 3.3. Exercise. Let U be anM-amenable ultrafilter on the parametricallyM-definable subsets of M . We begin by showing that U2 is a filter on the parametricallyM-definable subsets of M2. Let X,Y ⊆M2 be parametrically M-definable. Let Z = {m ∈M : (X ∩ Y )|m ∈ U}, Z1 = {m ∈M : X|m ∈ U}, Z2 = {m ∈M : Y |m ∈ U}. Since U is M-amenable, Z, Z1 and Z2 are parametrically M-definable subsets of M . Now, Z = {m ∈M : X|m ∩ Y |m ∈ U}. And, since U is a filter, X|m ∩ Y |m ∈ U if and only if X|m ∈ U and Y |m ∈ U . Therefore, using the fact that U is a filter, X ∩ Y ∈ U ⇐⇒ Z ∈ U ⇐⇒ Z1 ∩ Z2 ∈ U ⇐⇒ Z1 ∈ U and Z2 ∈ U ⇐⇒ X ∈ U and Y ∈ U. It follows that if X,Y ∈ U2, then X ∩ Y ∈ U2; and if X ∈ U2 and X ⊆ Y , then Y ∈ U2. Therefore U2 is a filter. We are left to verify that U2 is an ultrafilter. Let X and Z1 be as above. Let W = {m ∈M : (M\X)|m ∈ U}. Again, since U is M-amenable, W is a parametrically M-definable subset of M . Note that W = {m ∈M : M\(X|m) ∈ U}. And, since U is an ultrafilter, for all m ∈M , M\(X|m) ∈ U if and only if X|m / ∈ U . Therefore W = {m ∈M : X|m / ∈ U} = M\Z1.