It is and it isn't Purdue. It is a campus in the Purdue system. There are other campuses aside from the main one. For example, there are the Purdue Northwest Lions, a whole different university in the same system. Being admitted there does not get you into the Purdue everybody knows nationally. The chancellor is a different person and the accreditation is separate. Purdue Global is like that, not an arm of the main campus but its own campus in the Purdue system. But it is different from most state university systems in that I doubt most people know Purdue is a system, with multiple campuses. You expect that from "University of Texas" or "SUNY" but not from "Purdue." Being part of the Purdue system does mean something. I just think it's difficult to list it without confusing people. Or maybe irritating people who went to the main Purdue (Boilermakers) and know what that means. Just my opinion, though, which is worth what you paid for it.
