On Tue, Mar 5, 2019 at 8:03 AM Bruno Marchal <marc...@ulb.ac.be> wrote:
> *The expression "Non computable numbers” appears only in intuitionist > logic,* If so then just by reading the title of Turing's famous 1936 paper where he first described a device that we now call a Turing Machine you'd have to conclude that Turing was a intuitionist, it was called "On Computable Numbers, with an Application to the Entscheidungsproblem". John K Clark > On 4 Mar 2019, at 23:31, John Clark <johnkcl...@gmail.com> wrote: > > On Mon, Mar 4, 2019 at 11:04 AM Bruno Marchal <marc...@ulb.ac.be> wrote: > > >> I don't follow you. If the 8000th BB number is unknowable then it is >>> certainly uncomputable >> >> >> *> That is not true. All natural number n are computable. The program is >> “output n”.* >> > > I think you're being silly. You're saying if you already know that the > answer to a problem is n then you can write a program that will "compute" > the answer with just a "print n" command. But that's not computing that's > just printing. > > > I am even more silly. I claim that I need only to know that there is an > answer to say that BB(n), unlike [n]BB(n), is computable, trivially and non > interestingly, perhaps, but that follows from the classical definition of > Turing, Church, Markov, Hebrand-Gödel, etc. > > > > > > Incidentally very recently Stefan O’Rear has reduced Aaronson' s 7918 > number so now we know that BB(1919) is not computable. > > > Nice!!! > > Of course, we know only that BB(1919) = k, for k any enough big number is > undeciadbale in ZF. > > Perhaps tomorrow, we will know that BB(1919) = k is decidable in ZFC + > kappa. > > > > > So we know that: > • BB(1)=1 > • BB(2)=6 > • BB(3)=21 > • BB(4)=107 > > and that's all we know for sure, but we do know some lower bounds: > > • BB(5) ≥ 47,176,870 > • BB(6) ≥ 7.4 *10^36534 > • BB(7) >10^((10^10)^(10^10)^7) > > > *BB(n) is not computable means that there is no algorithm, which given >> n, will give BB(n).* >> > > Yes, so what are we arguing about? > > > > That we should not confuse the many possible notions of computable > functions from R to R, for which there is no standard definition on which > everyone would agree, and no corresponding Church-Turing notion, with the > notion of computable function from N to N (or any set of finitely > describable objects, always trivially computable). > > Mechanism use the Church-Turing notion. A digital brain has no real > numbers as input; nor real numbers as output. > > In the classical theory of computability, a real number is seen as an > infinite objects, and is modelled by total computable functions; or by > recursive operator, not by the usual partial recursive functions (phi_i). > > > > > > >> > *what Aaronson has shown, is that above 7918, we loss any hope to find >> it by using the theory ZF. But may be someone will find it by using ZF + >> kappa, which is much more powerful that ZF,* >> > > It's easy to find a system of axioms more powerful than ZF, the problem is > it may be so powerful it can even prove things that aren't true. > > > That is always the risk. It cannot been avoided. Provability is a relative > notion. No provers can prove its own consistency. > > > > Would you really trust a system that claimed to be able to solve the > Halting Problem? I certainly wouldn't! And if you can't solve the Halting > Problem then there is absolutely no way to calculate BB(7918) or BB(1919) > and I wouldn't be surprised if even BB(5) is out of reach. > > > > You are right on this. The BB function computability is equivalent with > computability with the halting oracle. > With an oracle for BB, you can solve the halting problem and vice versa. > > You can’t solve the totality/partiality problem though (named TOT). Even > with the BB oracle (or the halting oracle) you need to do an infinite task > to solve the TOT problem. There is a transfinite set of set of numbers > which are more and more unsolvable in that sense. > > > > > > > There are only 2 possibilities, a program will halt after a finite >>> number of steps or it won’t. >> >> >> > Yes. But the program which computes BB(n) always stop. >> > > if it stops then it is successful but if n is 1919 then it never stops so > BB(1919) is never computed. > > > AAronson’s paper is not on the computability of BB, but of the > undecidability of equation of the type BB(n) = k, in ZF. It is not about > us, but about the particular Löbian machine ZF. > > > > > >>I would maintain that you haven't solved a problem if you can't give >>> the right answer more often than random guessing would. >> >> >> *> You are restricting computability to a string notion of intutionistic >> computability. * >> > > What grade would you give a student who couldn't answer your questions > better than you'd expect from random guessing? > > > Random guessing has nothing to so with what we are talking about. > > My point is just that the notion of computability used is the classical > one, and in that case, despite [n]BBn is not a computable function, each > BB(n) is trivially, and non interestingly computable. That is why we use > natural numbers: to have a set of trivially computable primitive elements. > To compute any natural numbers, you need only to apply the successor > function s a finite number of time on 0. > > The expression "Non computable numbers” appears only in intuitionist > logic, or it means non computable real numbers. But real numbers are > (total) functions in disguise, and the notion of computability on real > numbers, or on more complex objects, admits many non equivalent > definitions. > > Bruno > > > > > 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 everything-list+unsubscr...@googlegroups.com. > To post to this group, send email to firstname.lastname@example.org. > Visit this group at https://groups.google.com/group/everything-list. > For more options, visit https://groups.google.com/d/optout. > > > -- > 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 post to this group, send email to email@example.com. > Visit this group at https://groups.google.com/group/everything-list. > For more options, visit https://groups.google.com/d/optout. > -- You received this message because you are subscribed to the Google Groups "Everything List" group. 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