On Thursday, June 4, 2020 at 6:07:45 AM UTC-5, Bruno Marchal wrote: > > > > On 2 Jun 2020, at 19:34, 'Brent Meeker' via Everything List < > everyth...@googlegroups.com <javascript:>> wrote: > > > > > > > > On 6/2/2020 2:49 AM, Bruno Marchal wrote: > >>> On 1 Jun 2020, at 22:43, 'Brent Meeker' via Everything List < > everyth...@googlegroups.com <javascript:>> wrote: > >>> > >>> > >>> > >>> On 6/1/2020 2:08 AM, Bruno Marchal wrote: > >>>> Brent suggest that we might recover completeness by restricting N to > a finite domain. That is correct, because all finite function are > computable, but then, we have incompleteness directly with respect to the > computable functions, even limited on finite but arbitrary domain. In fact, > that moves makes the computer simply vanishing, and it makes Mechanism not > even definable or expressible. > >>> That's going to come as a big shock to IBM stockholders. > >> > >> Why? On the contrary. IBM bets on universal machine > > > > No, they bet only on finite machines, and they will be very surprised to > hear that they have vanished. > > They bet on finite machines … including the universal machine, which I > insist is a finite machine. That is even the reason why I called it from > times to times universal number. > > I recall that once we get the phi_i, which can be defined in elementary > arithmetic, we get all the universal numbers, that is all u such that there > phi_u(x, y) = phi_x(y), and such u can be used to define all the recursive > enumeration of all digital machines. > > The implementation of this fine but universal machines are called > (physical) computer, and is the domain of expertise of IBM. > > Bruno > > Of course any computation is going to be finite or involve a finite number of bits. This happens as well with quantum computers, but there is one difference. Two states can be prepared and entangled so they have a continuum of probabilities depending upon measurement angle. This is what separates QM from classical mechanics. This separates entanglement of spins from the Bergman's socks, where knowing the left sock is in one box the right must be in the other. So while there is a finitude to the entanglement entropy or the quantity of quantum information, the possible ways an entanglement can register outcomes is infinite. This is what gives a violation of Bell's theorem in QM. With the measurement of a quantum system the pair of a state and measurement forms a type of Godel numbering. This connects QM foundations with the phi_u(x, y) = phi_x(y),you state above.
A classical computer will always be finite, and you can't have an infinite Cantor diagonalization. The computers that are manufactured are done so to solve certain problems, RSA encyrption, user interfaces for service personnel from travel agents to sales, word processors, games, cell phone signal shifters, data processors of medical measurements and on it goes. Even with quantum computers this will take off, and in fact I have thought quantum computing would be a way of managing a dynamics network defined by millions of drones over a city. Even if as I think the Godel-Turing result underlies obstructions between entanglement types quantum computers will in time become the province of engineering and business applications. LC > > > > > > Brent > > > >> and know well what is a computer: a finite arithmetical being in touch > with the infinite, and indeed, always asking for more memory, which is the > typical symptom of liberty/universality. IBM might be finitist, like > Mechanism, but is not ultrafinist at all. Anyway, mathematically, Mechanism > is consistent with ulrafinitsim, even if to prove this, you need to go > beyond finitism, (but then that’s the case for all consistent theory: none > can prove its own consistency once “rich enough” (= just Turing universal, > not “Löbian”). > >> > >> Bruno > > > > > > -- > > 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 everyth...@googlegroups.com <javascript:>. > > To view this discussion on the web visit > https://groups.google.com/d/msgid/everything-list/3068f558-7f61-56cb-61fe-44832ec28a91%40verizon.net. > > > > -- 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 everything-list+unsubscr...@googlegroups.com. To view this discussion on the web visit https://groups.google.com/d/msgid/everything-list/1cde9ecd-fc90-4031-a117-3e67a43af888o%40googlegroups.com.
