OK I ll try to sort through your latest post here Let me know if I misinterpret something First you specify a theory axiomatic system namely ZFC P P P omega exists Let s call that T0 for easy reference Now you argue that T0 proves the existence of a model let s call it M1 the 1 because it s not a model of T0 but of a new theory such that M1 satisfies ZFC P P omega is in M1 Now do you mean the real P omega or do you mean that M1 satisfies P omega exists that is that there s an element of M1 that M1 thinks is P omega I m guessing the latter that M1 satisfies P omega exists You say aleph2 is in the model and is less than or equal to the cardinality of P omega Again I m guessing you mean that this is in the model s opinion that is M1 satisfies the statement exists and OK so going with my guesses thus far I agree that T0 proves the existence of such an M1 M1 might be for example M0 G where M0 equals which is a model for T0 and G is M0 generic for the partial order to add Cohen reals