 This multiverse conception is clearly related to the algebraic conception discussed in sections 3.1–3.2. Further, it satisfies one of the key desiderata that we isolated in section 3.2.  Hamkins defends the multiverse as an independently plausible conception of set theory, and he argues that the motivation for accepting it comes from within mathematical practice.  (I.e., Hamkins does not argue that, because forcing extensions are possible, we are stuck with set-theoretic relativity; rather, he argues that, because forcing extensions are natural, we should embrace set-theoretic relativity.)  In this sense, something like the multiverse could well constitute the “right” way of developing the algebraic conception.