Dear Lawrence,
as far as i understood it, Loeb's theorem says exactly what you wrote in the post above. This result indicates for me two things, firstly that there is no TOE which can be proven to be the 'real thing'. Because if one could prove it, it would be inconsistent and therefore wouldn't be the real thing, and therefore not the TOE.
Secondly, if mathematics has these malicious properties as Loeb's results indicate, then, for the sake of consistency, we must differentiate between provability and truth. This is what naturally all authors in the essay contest not do: although their lines of reasoning cannot be proven, they assume them to be nonetheless the truth (including my essay).
Claiming that one's result is the Truth in the absence of a proof, because these results appear to be so self-evident to the proponent would mean that the proponent equals self-evidence with a formal proof. But these both are different things. Self-evidence refers to the consistency of a certain line of reasoning, but does not say anything about the ontological status of its contents.
Now let's make a more general point: If mathematics would indeed be the fundamental layer of reality in a platonic sense, it would obey Loeb's theorem. Since all of mathematics then resides in the platonic realm, it must be complete. Every new axiom, identified by human beings, would not be a human creation, but a discovery of a part of that platonic realm. But this cannot be the case, since in the platonic realm, mathematics must be considered as complete (and infinitely infinite). But if it is complete, every sentence that can be constructed could be proven. But this implies that this mathematics is inconsistent.
Taking this scenario at face value, one then can return to the initial assumption and ask where the error lies. Does we find the error within Loeb's theorem or within Gödel's theorems? Or is it really true that mathematics does not encompass all of reality, even if this assumption cannot be proven to be true? I think the latter is the most probable answer: Mathematics cannot be the most fundamental level of reality, because otherwise we run into contradictions within our own lines of reasoning.
If something such rational and calculatable as mathematics should not be the most fundamental level of reality, what then should be this level? I have argued in my essay that it needs an intelligent entity who at least invented mathematics. Otherwise one had to conclude that reality is an absurdity, producing or providing a system (mathematics) that mimics some rational and consistent behaviour but nonetheless, at its core, it must have arisen out of a sheer inconsistency, a kind of absurd nothing. Fortunately the latter can also not be proven and if one assumes it to be nonetheless true, how can one then be sure that even Loeb's theorem tells us something meaningful about reality?
I think from a logical point of view one has to cope with the fact that mathematics has certain limits, limits which are a broad hint that mathematics cannot be the most fundamental level of reality - because in an inconsistent reality, the very notion of 'fundamental' may not carry any sense of ontology with it. For the existence of a most fundamental level of reality one could expect that it shows up from time to time in a manner that contrasts the widely held assumption of the omnipotence of mathematics (as i tried to show by the example of near-death experiences). Since we are not able to solve some 'simple' tasks like the 3- or 4-body problem and other physical tasks, the assumed omnipotence of mathematics seems not to be fully implemented at least in our physical universe. And if it nonetheless would, this 'omnipotence' necessarily would lead to inconsistencies due to Loeb's theorem. But this would lead us again to our initial assumption of how we then can validate the soundness of all of mathematics itself, including Loeb's and Gödel's theorems. As Loeb indicated, we can't do this, even in a world where the assumed omnipotence of mathematics is physically instantiated. Thus, the assumed omnipotence of mathematics is only assumed, but can never be reached, neither in a platonic realm nor in a physical realm, because incompleteness and inconsistence are mutually excluding each other. Therefore, neither an incomplete nor an inconsistent system is a good candidate for the most fundamental level of reality. By pondering about the alternative, i think one has simply to assume a teleological component behind it all (without ever being able to prove this mathematically).
What do you think about these lines of reasoning?
Best wishes,
Stefan Weckbach