 One thing to note here is that Hamkins is open to the idea that some universes may be models of theories other than ZFC (weaker theories such as Zermelo set theory or Peano arithmetic, or even different theories such as ZFA or NF/NFU). But it appears to be difficult philosophically to articulate exact boundaries for what counts as a “concept of set theory” which would define a universe. And this is fine, because there is no notion of “the multiverse” of universes as a completed totality here at all—this would amount to smuggling in the single Platonic universe again through the back door! Some of the axioms which follow do presume that the universes discussed are models of ZFC or very similar theories.