Your note about Gödel’s theorem is confusing or doesn’t make sense. There is no such thing as an inconsistent math structure, assuming that by “structure” you mean the things used in defining the semantics of first order logic (which is what Tegmark means when he says “structure”, unless I’m mistaken).
The incompleteness theorems only give limitations on recursively enumerable sets of axioms.
Other than that, this looks like a great resource for people wanting to investigate the topic for themselves.
I would also add just to remember the idea, that logical paradoxes inside logical universe may look like logical black holes, and properties of these black holes may have surprising similarities with actual black holes.
Logical black holes may attract lines of reasonings, but nothing could come out of them, and in the middle they have something where main laws contradict each other the same way as physical laws are undefined in the gravitational singularity of astronomical black hole.
Epistemic status: crazy idea.
Thanks, I am not very kin with Godel theorem, but some paradoxes in math exists, like the one about set of all sets—does it contains itself? If we claim that math is final reality, we must find the way to deal with them.