I am not following this thread any more. I noticed Bruno writing about Gödel's theorem and mechanism. The role of such is hard to nail down. My take on the role of undecidability theorems in physics is below.
There is a famous story that John Wheeler asked Kurt Gödel whether his incompleteness theorems played any role in quantum mechanics, where upon Gödel threw Wheeler out of his office. Most usual answers about whether Gödel's theorem plays a role in physics is met with negativity by both mathematicians and physicists. Gödel's theorems are about a mathematical system with the power of first order logic or that is recursive not being able to make a complete list of all provable propositions about itself. Gödel showed this by illustrating how a free variable in a predicate can be the Gödel number for that predicate, which as itself a Gödel number is outside a list of such number in the manner of Cantor's diagonalization. This is Gödel's first theorem. The second theorem says that if there are unprovable theorems, which tacitly state their unprovability they must be true. This is because if they are false it leads to a contradiction that they are provable. However, the first theorem is most important here. This would pertain to physics with the question of whether physics can ever represent itself within itself. This runs into some issues, where Godel's theorem and the Cantor diagonalization implies an axiomatic system that has an infinite number of propositions. If we consider a proposition as some set of bits or quantum bits, then clearly what is accessible to any observer is finite. In fact the universe appears to conspire mightily to prevent any observer from accessing an infinite amount of anything. Singularities are hidden behind event horizons, even if the observable universe is infinite its expands in a way that leaves only a finite portion observable. We may consider the issue of hypercomputation. A trivial case of this is a switch that flips every halving of each interval of time, so if the initial interval is a second then an infinite number of switch flips occur in the next second. Malement-Hogarth spacetime have properties with Cauchy horizons that pile up incoming signals. It is then possible for an infinite computation to occur that can be accessed by an observer in a finite time. This hypercomputation permits a machine to compute beyond the limits of the Church-Turing thesis. However, these spacetimes may be pathological. In the case of the infinite flipping of the switch, the asymptotic divergence of the frequency of switch flipping means the ultimate limit is energy large enough to cause the switch to become a black hole. The inner horizon of a Kerr or Reisner-Nordstrom blackhole is continuous with I^+, so an infinite number of null rays pile up there in a Cauchy horizon. This could be a set up for a hypercomputations. However, Hawking radiation decays the black hole so it is not eternal. Nature appears to conspire to prevent any circumvention of the Church-Turing thesis. This is exactly what we might expect if Gödel's theorem plays a role in nature. The natural physics to look at is quantum mechanics, where the observer is ultimately a quantum system that is accessing information about a quantum system. The observer plus system is a whole system that is in this setting self-referential. Recent work with the Frauchiger-Renner theorem and measurements related to the Wigner's friend have demonstrated limits on the capacity of quantum mechanics to access information about itself. This may have ultimately consequences for issues involving the foundations of quantum mechanics, in particular the Born rule, and with cosmological issues for how the universe is structured so that information physics conforms to the Church-Turing thesis. LC On Tuesday, April 6, 2021 at 10:32:57 AM UTC-5 [email protected] wrote: > On Tue, Apr 6, 2021 at 11:05 AM Bruno Marchal <[email protected]> wrote: > > *> I use the term “pagan” for “non confessional theology”* > > > Since you were the only one in the world who knows Brunospeak rather than > telling us what the synonyms in your homemade language are it might be more > interesting to all of us who don't speak that tongue if you would try > doing some actual philosophy instead. On second thought that's probably not > feasible because to do that you would be required to actually have > something to say. > > John K Clark > > -- You received this message because you are subscribed to the Google Groups "Everything List" group. To unsubscribe from this group and stop receiving emails from it, send an email to [email protected]. To view this discussion on the web visit https://groups.google.com/d/msgid/everything-list/75fd0d72-f881-4d9b-8f0a-56f37f17de69n%40googlegroups.com.
