On Wed, Jul 5, 2017 at 11:51 AM, Bruno Marchal <[email protected]> wrote:
> >> >> the study of the largest prime number seems rather silly to me. > > > > > It is just a logical point that you can't derive this from Robinson > Arithmetic theory, > That doesn't bode well for the future prospects of the Robinson Computer Hardware Corporation now does it. > > > a bit like the Euclid postulate on parallel has been shown non derivable > from the other geometric axiom. > But neither Euclid 's Fifth P ostulate nor its absences leads to a world where logical contradictions can exist, but a finite number that is the largest prime does. >> >> A Turing machine can do Peano Arithmetic >> too. Nothing new in that. >> > > > > Of course not > , > Of course not ?? > > > I said that I wrote different program for a Löbian machine, and if you > read Smullyan's book, you will find others, and they are all emulable by a > Turing machine > If it can emulate it then whatever a " Löbian machine " is and whatever it can do a Turing machine can do it too, including Peano Arithmetic. As I said, nothing new. > > > you seem to ignore the difference between a theory, or a set of "believed" > propositions/sentences. > > A (universal) Turing machine can compute (everything computable), but > cannot prove anything. > As far back as 1956 a computer had proved 38 theorems in Whitehead and Russell's Principia Mathematica, and some of the proofs were judged by most to be more elegant than the proofs Russell and Whitehead found. And the old machine used to do it was equivalent to a Turing Machine, as are modern computers. >> >> >> Turing did far more than define what his machine could do, he explained >> exactly how to construct one in great detail, and that's why other people >> gave it the name "Turing Machine". > > > > > Yes, he is the discoverer of the (mathematical) notion of computer, and he > was interested in building one. > It's true Turing built actually electronic devices but what I meant was in in his original article about a long paper tape and marking pen and a eraser etc he described exactly how to build one; he didn't expect anyone would actually build a practical machine that way but he did prove it was physically possible to do so. Where is your equivalent for a Löbian machine? Martin Löb knew exactly what is a Löbian machine, although he would have > called it a "sufficiently rich theory" I know what departments in my big box hardware store to go to to buy light detectors, paper tape, erasers, marking pens, and gears and pulleys to build my Turing Machine; but what department sells "sufficiently rich theory" and how heavy is it, will I need help carrying it to my car? > > Like I said, PA cannot prove its own consistency, but PA can prove that > ZF can prove PA's consistency, > That doesn't solve the problem, it just kicks the problem down the road. I s Zermelo–Fraenkel consistent? ZF can't say, and even if it is there are true statements that is can't prove. 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 post to this group, send email to [email protected]. Visit this group at https://groups.google.com/group/everything-list. For more options, visit https://groups.google.com/d/optout.
