On Fri, May 17, 2019 at 10:47:36PM +1000, Bruce Kellett wrote: > On Fri, May 17, 2019 at 10:14 PM Bruno Marchal <marc...@ulb.ac.be> wrote: > > On 16 May 2019, at 03:27, Bruce Kellett <bhkellet...@gmail.com> wrote: > > On Thu, May 16, 2019 at 12:59 AM Bruno Marchal <marc...@ulb.ac.be> > 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, acceptance of the Church-Turing thesis (ie the existence of a universal Turing machine) 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). 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. 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. Bruno's work does not address this ultrafinitist case, as the CT thesis is an explicit assumption. Except that the Movie Graph Argument is supposedly about that case. Cheers -- ---------------------------------------------------------------------------- Dr Russell Standish Phone 0425 253119 (mobile) Principal, High Performance Coders Visiting Senior Research Fellow hpco...@hpcoders.com.au 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 everything-list+unsubscr...@googlegroups.com. To view this discussion on the web visit https://groups.google.com/d/msgid/everything-list/20190519002736.GK5592%40zen.
