In his paper on mathematical universe hypothesis, Max Tegmark only responses with a single paragraph to this assumption:
The MUH and the Level IV multiverse idea does certainly not imply that all imaginable universes exist. We humans can imagine many things that are mathematically undefined and hence do not correspond to mathematical structures. Mathematicians publish papers with existence proofs and demonstrating the mathematical consistency of various mathematical structures precisely because this is difficult and not possible in all cases.
To me, this argumentation sounds to me like the appeal to ridicule. Personally, I think the theory does imply all imaginable universes exist and thus even those where dragons are presidents if such universes can be logically consistent and the fact such imaginations seem ridiculous doesn't make them invalid. My reasoning for the assumption every possible universe exists stems from a notion Tegmark himself acknowledges:
Stephen Hawking famously asked “what is it that breathes fire into the equations and makes a universe for them to describe?” . In the context of the MUH, there is thus no breathing required, since the point is not that a mathematical structure describes a universe, but that it is a universe.
My understanding of this statement is - there's no reason for a mathematical (=abstract) structure not to exist. That's because there's no "higher logic" which could "dispute" any mathematical system, logic itself is a product of these systems. Thus, even when we formulate a question like "Where does it come from?" we're using the logic which is a product of our universe and doesn't make any sense outside of it.
Have I misunderstood Tegmark or is there any background reasoning that supports his claim MUH does not imply all imaginable universes exist?