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    $\begingroup$ Alec, your post (and title) consistently uses the phrase "the multiverse". I see the mathematical value of having an individual constant symbol, but why do you call it "the multiverse" instead of "a multiverse"? (I ask because it strikes me as a little strange to allow for pluralism at the level of "class of all sets", but not at the next level.) $\endgroup$ Commented May 24, 2022 at 18:48
  • $\begingroup$ @PaceNielsen Good point; implicitly I am hoping that the 'correct' notion of multiverse will admit a metatheorem to the effect that any 'multi-multiverse' we define is already equiconsistent with the 'correct' theory of the multiverse. But I have no idea if such a metatheorem holds for any definition, let alone this one, so I take your point and I'll edit accordingly. $\endgroup$ Commented May 25, 2022 at 18:25
  • $\begingroup$ @PaceNielsen When trying to edit just now, I realized that 'the multiverse' I am referring to is the one that Hamkins asserts set theorists implicitly work in when considering forcing arguments and the like. For me to say that I am axiomatizing 'a multiverse' makes the question pointless in a certain sense; basically whatever I consider to be 'a multiverse' would be an affirmative answer, but that's a boring question. I suppose what I mean by asking for 'the multiverse' is the one implicitly used by set theorists, and wether or not these axioms fully and faithfully describe that multiverse. $\endgroup$ Commented May 25, 2022 at 18:32
  • $\begingroup$ Hi Alec, Could you possibly elaborate on how your proposed axiomatisation is different from Hamkins's, and what you are trying to accomplish in non-technical terms? $\endgroup$ Commented Jun 22, 2023 at 16:34
  • $\begingroup$ I'm having a bit of difficulty seeing how axiom 8 is going to work. First of all, it's not really clear how you can formalize the notion of an object's existence being instantiated by an axiom without putting in explicit machinery to track this (e.g., Skolem functions). Also, if $V \preceq V+\phi$, then $V$ and $V+\phi$ have the same first-order theory. This means that if $\phi$ is a first-order property of the $V$ in question, you can't have both $V+\phi$ and $V+\neg \phi$ existing as elementary extensions. $\endgroup$ Commented Jun 22, 2023 at 16:58