> On 3 Jun 2021, at 13:28, Lawrence Crowell <[email protected]> > wrote: > > It is Penrose's thesis that consciousness is a sort of Godel trick.
? Penrose on the contrary use Gödel’s theorem, like Lucas, erroneously, to claim that we are not machine. Basically he says that the correct machine cannot see []p -> p, but that we can see it, even for us. But he just confused G and G*. In other place they confuse G and S4Grz. > Back in the 1980s as an undergraduate I would have agreed with this, when I > started reading about this. I read Hofstadter's book "Godel, Escher, Bach" > and began pondering these things. I have however come to think there were > problems with this. It is clear humans are not consistent Turing machines or > computers. That is a consequence of mechanism. To be consistent at some level, the “variable” machine have to develop a non monotonically layer, and yes, you are taking risk when saying “yes” to quick to the charlatan-doctor... > Computers are infernally consistent, Before you install windows, I guess :) A computer is just a relative universal number, it can imitate (and sometimes even become) all other digital machines, both the consistent and the inconsistent one. To be precise: consistency apply to theories, that is set of beliefs, or assertable or provable propositions. Computability is an absolute notion. Provability is a relative notions. Computation, translate into provability, is sigma_1 provability, already entirely obtained by the jewel Q: Classical First Order Logic + Equality, +: 1) 0 ≠ s(x) 2) x ≠ y -> s(x) ≠ s(y) 3) x ≠ 0 -> Ey(x = s(y)) 4) x+0 = x 5) x+s(y) = s(x+y) 6) x*0=0 7) x*s(y)=(x*y)+x Which, BTW, is among my favorite theory of everything (ontology). I insist that theology, and thus physics, does not depend on the choice on the ontology, as long as we avoid induction axioms, and the axiom of infinity (in that sense, Mechanism is quite atheistic no creator, no creation, just the dream of number entailed by the Turing universal/complete theory Q. My second favourite theory of everything is the theory of combinator CL (I explained it up to its Turing universality some years ago): Axioms: KAB = A SABC = AC(BC) Inference rules: If A = B and A = C, then B = C If A = B then AC = BC If A = B then CA = CB The laws of physics can be said to emerge, in an Emmy-Notherian generalised way, from the invariance of the observable for all universal numbers. > and can compute numerical sequences, but they do not make an inductive leap > in saying the set of natural numbers has infinite cardinality. The Löbian universal machine known as ZF does that all the time. You might mean that she does not learn to do that? Wait for alpha-go learning a bit of set theory. They might make an inductive leap that the humans will take some times to understand, at some point. And only God will know if such machines are consistent or not. Are you telling me that you would always say “no” to the doctor (for the artificial brain?). I don’t know the truth, but when you listen to the machines, through Gödel, Löb … Solovay, you understand that the universal machine are born in arithmetic, are right at the start confronted to an hesitation between Security (totality, control) and Insecurity, like searching for some numbers which might, or notion exists, going from surprise to surprise… They are never completely satisfy and want always more, until they wake up, to fall asleep again... > Humans can rather easily see the set is infinite and however make mistakes. When machine do inductive inference, or pattern recognition, or play chess, or whatever, they do mistakes. They don’t do mistake at the level of their implementation, but you don’t do that either; you just don’t mess with the physical laws, nor machine mess with the arithmetical laws which implements them. Hofstadter (the only physicist who get Gödel’s right) got this right, and sum it well up with the image of a lot if 1+1=2 building a picture I-of 1+1=3. You can put Gödel’s theorem in this form: (note that <>t is the same as ~[]f, consistency) <>t -> ~[]<>t That is; a consistent machine/entity cannot prove its consistency. But that is equivalent to <>t -> <>[]f For a consistent machine, it is consistent to be(come) inconsistent. And indeed, PA + ~con(PA) is consistent, and that is already a sort of axiom of infinity making that unsound, but still consistent, machine, far more powerful than PA. Arithmetic is full of life, but you get the many liars as an uncomfortable but unavoidable gifts… Bruno > > LC > > On Wednesday, June 2, 2021 at 11:47:26 AM UTC-5 [email protected] wrote: > On Tue, Jun 1, 2021 at 8:38 AM Lawrence Crowell <[email protected] > <applewebdata://EB8C8B7C-057F-4507-B793-DA4C2DE4F4CB>> wrote: > > Godel's theorems are our friend. It is even a friend in physics. With > > physics I think it is a "sieve" that conforms physical principle to have > > horizon conditions, whether uncertainty principles or event horizons in GR, > > that conform physical reality to fit within the Church-Turing thesis. > > Some claim Godel proved that the human mind is more than just a Turing > Machine, but I disagree. Godel found a way to use numbers to write a sentence > that talks about itself, it says "I am not provable in this formal system", > and the operations of a particular Turing Machine are analogous to a formal > system; however a human being can look at that sentence and see that it is > true even though the machine itself could never produce it, therefore the > human mind can do something the Turing machine can't. However, what Godel > proved is that an operating system powerful enough to perform arithmetic THAT > IS CONSISTENT cannot be complete, and he says no operating system can prove > its own consistency. But when human beings are not doing formal logic > exercises but just living everyday lives their operating system is most > certainly not consistent, they can have two logically contradictory opinions > at the same time, a brief glance at politics shows it is very common. And > humans can be absolutely positively 100% certain about something, (that is to > say they have proven it to their own satisfaction), and still be dead wrong. > Godel's biography illustrates this point, he refused to eat and died of > starvation because he was absolutely positively 100% certain that his food > was being poisoned. > > So we are inconsistent Turing machines. And even today we could easily make > a machine that could answer any question, provided you don't mind if it > sometimes gave an answer that was wrong or even idiotic. > > 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] > <mailto:[email protected]>. > To view this discussion on the web visit > https://groups.google.com/d/msgid/everything-list/3121aa56-24ca-459e-a196-a31a960d356bn%40googlegroups.com > > <https://groups.google.com/d/msgid/everything-list/3121aa56-24ca-459e-a196-a31a960d356bn%40googlegroups.com?utm_medium=email&utm_source=footer>. -- 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/2CEA239F-55D1-4E7D-AD82-ABD4EE0ADFD7%40ulb.ac.be.
