Hi Lawrence, thanks for your excellent comments and your thoughts envoled in them. I will give my answers subsequently in this reply according to the points you made.
According to Russell's barber, this paradox necessarily presupposes that *none* of the other men (wich means the group of men in the village *except* the barber) are allowed to shave those other men of the village. Otherwise the barber wouldn't be a barber - *according to Russell's definition*. So, Russell's whole definition of a barber is *ill-defined*, since this is not what a barber must be defined, even for the case that there is only one barber in the village. The point is, that Russell's definition of the barber as "no men in the village are allowed to shave other men - except the barber" (this is the real fixed-point that can be extracted from the context in which Russell's definition seems to be true) does *not imply* that people are *not* allowed to shave themselves. Therefore the barber can shave himself.
What you say about Gödel's theorem and provability is interesting. It all crucially relies on whether or not logics is able to capture some fundamental truths. If logic and with it the mathematical systems which produced Gödel's results in the first place are inconsistent, then of course everything is provable with the help of these systems. So, the presupposition that logics is consistent demands that there are some true statements within those systems which are not provable with the help of these systems.
Otherwise one potentially could prove a system to be inconsistent and incomplete, but could also prove that a system is *not* inconsistent and incomplete. So would the latter proof be a stronger one, in the sense that the system under consideration should be considered as consistent, but incomplete? This is a senseless question, since we observed from the very start that the system generated a contradiction, leading to *everything* at the end of the day. If you can't make anymore a reliable distinction with a certain system, it is then senseless to further use this system.
Gödel's results are only *fundamentally* true under the following two presuppositions:
1. logics is consistent
2. Mathematics is eternal and infinite
If one of these two presuppositions is false, Gödel's results have no fundamental impact whatsoever. In my essay I argue that the second point may be false in the sense that our traditional view of mathematics as an eternal platonic realm is difficult to reconsile with Gödel's results, since every extension of a mathematical system critically hinges on what one considers to be a necessary additional axiom - for making such an extension not only *consistent*, but *eternally true*.
The problem is, if there exists an infinite, non-denumerable number of truths within this mathematical landscape, then also Moore's theorem should hold and every physically or mathematically defined final theory of everything would be a final theory of nothing - when refered to the question why our universe is what it is - and why mathematics is what it is. The suggested advantage of an eternal landscape of mathematics is that it seems to justify that there exists something at all, rather than nothing. What brings me back to the very beginning, since even the quest for the existence of God is in some form an instance of unprovability - in the same sense the eternal mathematical landscape is. In this sense, people knew all along, long before Gödel's results, that there may be things which cannot be logically proven nor disproven, but nonetheless could be true. Surely, the mathematical universe hypothesis, for example, does rest on empirically gathered data, means, on the truth that nature indeed incorporates a certain amount of mathematics. But it is also true that it incorporates a certain amount of consciousness. Max's (the latter I appreciate in very many respects) claim of the MUH rests (beneath others) on the assumption that even consciousness is fully formalizable by mathematics. I doubt this by saying that mathematics can at the maximum merely establish correlations between some brain actions and some mathematical patterns.
The question of how to properly interpret such correlations is a fundamental one. I am currently working on this, but cannot present yet any robust results. Trying to describe a strictly deterministic system in terms of axioms seems to be impossible to me other than taking it at face value and therefore as a true axiom that all that exists is indeed 'merely' a strictly deterministically acting system. It could turn out that a superposition of states yields *less* information about the system then the parts of the system themselves. Therefore it is crucial for me to look how one can incorporate an observer into quantum mechanics, the latter being independent of a strict determinism, but nonetheless being able to have some limited free-will at hand to decide between two mutually exclusive options. And you are right about closed systems. As I described in my essay, it may turn out that defining ultimate reality solely in terms of formal systems may itself be just a closed system. I am working on stepping out of such a system in a logically and meaningful manner in terms of how to properly interpret a global wave function. If I succeed, I surely will publish what I found.
