> On 19 May 2019, at 02:27, Russell Standish <[email protected]> wrote: > > On Fri, May 17, 2019 at 10:47:36PM +1000, Bruce Kellett wrote: >> On Fri, May 17, 2019 at 10:14 PM Bruno Marchal <[email protected]> wrote: >> >> On 16 May 2019, at 03:27, Bruce Kellett <[email protected]> wrote: >> >> On Thu, May 16, 2019 at 12:59 AM Bruno Marchal <[email protected]> >> wrote: >> >> The first order theory of the real numbers does not require >> arithmetical realism, but the same theory + the trigonometrical >> functions reintroduce the need of being realist on the integers. >> Sin(2Pix) = 0 defines the integers in that theory. >> >> If you reject arithmetical realism, you need to tell us which >> axioms you reject among, >> >> 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 >> >> >> You say that "realism" is just acceptance of the axioms of arithmetic above. >> But then you say that arithmetical statements are true in the model of >> arithmetic given by the natural integers. There is a problem here: are the >> integers the model of your axioms above, or is it only the axioms that are >> "real". If the integers are the model, then they must exist independently of >> the axioms -- they are separately existing entities that satisfy the axioms, >> and their existence cannot then be a consequence of the axioms, on pain of >> vicious circularity. > > > Axioms 1-3 define the successor operator s(x). It is enough to > generate the set of whole numbers by repeated application on the > element 0. As a shorthand, we can use traditional decimal notation (eg > 5) to refer to the element s(s(s(s(s(0))))). 4&5 define addition, and > 6&7 define multiplication on these objects. > > Goedel's incompleteness theorem demonstrates there are true statements > of these objects that cannot be proven from those axioms alone. > > In that sense, the whole numbers are a consequence of those axioms, > whilst also being separately existing entities (having a life of their own). > > There are also nonstandard airthmetics, that involve adding additional > elements (infinite ones) that cannot be created by successive > application of s. > > Given these 7 axioms can also be viewed as an algorithm for generating > the whole numbers,
And generating all computations. That is proved by showing that all partial computable function are represented in that theory. If phi_i(j) = k, that can be proved in that theory (amazingly enough). > acceptance of the Church-Turing thesis (ie the > existence of a universal Turing machine) Well, I will cut the hair, but of course, the existence of the universal Turing machine is a theorem in that theory. Church thesis is the thesis that this Tiuring universal machine is truly universal for the intuitive notion of computability. > is sufficient to reify the > whole numbers. Conversely, this arithmetic is sufficient to generate > all possible Turing machine (IIRC, the proof involves Diophantine > equations, but wiser heads then me may confirm or deny). Very simple one. Like defining x < y by x = y + a (or Ea(x = y + a). It happens that the Diophantine equations are already Turing universal, but that is far difficult to show. Cf the works of Putnam, Davis, Robinson, and finally Matiyasevich. > > A converse position (held by a small minority of mathematicians) is > that perhaps not all whole numbers exist - that there is some > (unspecified) maximum integer x for which s(x) is not meaningful, and > in particular, for which axiom 3 is false. ? 3 just say that all successor have a predecessor. It remains true for ultrafinitism. Contrary to what I claimed a long time ago, ultrafinitim is consistent with mechanism. It is just weird, but it i.e. easy to build a model of the theory above with a bigger natural number, like it is easy to build a model of the theory above in which 0 + x is different from x + 0. The theory is very weak, yet Turing complete. > In such an environment, the > CT thesis must be false, there can be no universal machine capable of > emulating all other others - there must be at least one such machine > whose emulation program is too long to fit on the obviously finite length > tape. This does not follow. > > Bruno's work does not address this ultrafinitist case, as the CT > thesis is an explicit assumption. CT just looks weird with ultrafinitism, and we would lose the notion of Löbianity, but we can still study the Löbian machine in the ultrafinitist position. The physical universe would just be even more delusional, but not much more than with finitism. > Except that the Movie Graph Argument > is supposedly about that case. I don’t understand this. Bruno > > Cheers > > -- > > ---------------------------------------------------------------------------- > Dr Russell Standish Phone 0425 253119 (mobile) > Principal, High Performance Coders > Visiting Senior Research Fellow [email protected] > Economics, Kingston University http://www.hpcoders.com.au > ---------------------------------------------------------------------------- > > -- > 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/20190519002736.GK5592%40zen. -- 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/333BAEA9-383A-4587-B650-FA7C190B4D9B%40ulb.ac.be.
