A model in laymen's terms is a mathematical structure (sets when within set theory) that satisfies a theory (or alternatively a set of axioms). For example the model Vω is a model of a theory of purely finite sets, sets like the set of all natural numbers N cannot be exist in this model (In what sense are inaccessible cardinals inaccessible? - second answer). What if we took the universe of sets VΩ (defined as the union of all of the cumulative hierarchy V1 ⋃ V2...) as a model for a theory (e.g. ZFC)? What are the implications of this? What would this theory be like? Is this even possible? I ask this because it appears we would have a theory that proves ALL sets (including inaccessible and large cardinals as from a Platonist point of view they are sets). I have only recently begun researching 'models' and 'consistency' so the entire idea may introduce several problems and paradoxes I am unaware of (please feel free to inform me if so).
As for clarification, since a theory is based off it's axioms, the theory ZFC (or a theory which has the cumulative hierarchy) will do for this example.