So, let’s suppose for a moment that ZFC set theory is the one true foundation of mathematics, and it has a “standard model” that we can meaningfully point at, and the question is whether our universe is somewhere in the standard model (or, rather, “perfectly described” by some element of the standard model, whatever that means).
In this case it’s easy to imagine that the universe is actually some structure not in the standard model (such as the standard model itself, or the truth predicate for ZFC; something along those lines).
Now, granted, the whole point of moving from some particular system like that to the more general hypothesis “the universe is mathematical” is to capture such cases. However, the notion of “mathematics in general” or “described by some formal system” or whatever is sufficiently murky that there could still be an analogous problem—EG, suppose there’s a formal system which describes the entire activity of human mathematics. Then “the real universe” could be some object outside the domain of that formal system, EG, the truth predicate for that formal system, the intended ‘standard model’ of that system, etc.
I’m not confident that we should think that way, but it’s a salient possibility.
So, let’s suppose for a moment that ZFC set theory is the one true foundation of mathematics, and it has a “standard model” that we can meaningfully point at, and the question is whether our universe is somewhere in the standard model (or, rather, “perfectly described” by some element of the standard model, whatever that means).
In this case it’s easy to imagine that the universe is actually some structure not in the standard model (such as the standard model itself, or the truth predicate for ZFC; something along those lines).
Now, granted, the whole point of moving from some particular system like that to the more general hypothesis “the universe is mathematical” is to capture such cases. However, the notion of “mathematics in general” or “described by some formal system” or whatever is sufficiently murky that there could still be an analogous problem—EG, suppose there’s a formal system which describes the entire activity of human mathematics. Then “the real universe” could be some object outside the domain of that formal system, EG, the truth predicate for that formal system, the intended ‘standard model’ of that system, etc.
I’m not confident that we should think that way, but it’s a salient possibility.