Tegmark extends this idea to hypothesize that all mathematical objects exist, which he describes as a form of Platonism or Modal realism.
Jürgen Schmidhuber argues that it is not possible to assign an equal weight or probability to all mathematical objects a priori due to there being infinitely many of them.
Physicists Piet Hut and Mark Alford have suggested that the idea is incompatible with Gödel's first incompleteness theorem.
Andreas Albrecht when at Imperial College in London called it a "provocative" solution to one of the central problems facing physics.
Tegmark's response[10]: sec VI.A.1 is to offer a new hypothesis "that only Gödel-complete (fully decidable) mathematical structures have physical existence.
This drastically shrinks the Level IV multiverse, essentially placing an upper limit on complexity, and may have the attractive side effect of explaining the relative simplicity of our universe."
Tegmark goes on to note that although conventional theories in physics are Gödel-undecidable, the actual mathematical structure describing our world could still be Gödel-complete, and "could in principle contain observers capable of thinking about Gödel-incomplete mathematics, just as finite-state digital computers can prove certain theorems about Gödel-incomplete formal systems like Peano arithmetic."
Tegmark maintains that MUH is testable, stating that it predicts (a) that "physics research will uncover mathematical regularities in nature", and (b) by assuming that we occupy a typical member of the multiverse of mathematical structures, one could "start testing multiverse predictions by assessing how typical our universe is".[3]: sec.
[3]: sec V.C However, Jannes[13] argues that "mathematics is at least in part a human construction", on the basis that if it is an external reality, then it should be found in some other animals as well: "Tegmark argues that, if we want to give a complete description of reality, then we will need a language independent of us humans, understandable for non-human sentient entities, such as aliens and future supercomputers".
Brian Greene argues similarly:[14]: 299 "The deepest description of the universe should not require concepts whose meaning relies on human experience or interpretation.
However, there are many non-human entities, plenty of which are intelligent, and many of which can apprehend, memorise, compare and even approximately add numerical quantities.
VI.A that math is evolving over time, there is "no reason to think it is converging to a definite structure, with fixed questions and established ways to address them", and also that "The Radical Platonist position is just another metaphysical theory like solipsism...
V.E that "This is less inconsistent with Level IV than it may sound, since many mathematical structures decompose into unrelated substructures, and separate ones can be unified."