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 t for easy reference now you argue that t proves the existence of a model let's call it m the because it's not a model of t but of a new theory such that m satisfies zfc p p omega is in m now do you mean the real p omega or do you mean that m satisfies p omega exists that is that there's an element of m that m thinks is p omega i'm guessing the latter that m satisfies p omega exists you say aleph 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 m satisfies the statement exists and ok so going with my guesses thus far i agree that t proves the existence of such an m m might be for example m g where m equals which is a model for t and g is m generic for the partial order to add cohen reals