On Sun, May 19, 2019 at 10:27 AM 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. > That is where the problem lies. If these axioms generate the set of whole numbers, then that is a constructvist or nominalist account of arithmetic. If, however, the integers exist independently and are thus just a model for these axioms (a domain in which the axioms are true), then you have arithmetic realism. You can't have it both ways. 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). > That is an independent assumption, not implied by the axioms above, as I have pointed out. > 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. That remains to be proved. Church-Turing is about calculable numbers, not about reification. It also works in a purely nominalist account. > 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. > Interesting, but not my immediate concern. Which is that the axioms and CT do not imply arithmetical realism: that has to be a separate assumption, and there is no independent justification for such an assumption. Bruce > 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 > -- 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/CAFxXSLQhexHK7kaEMYxS0Bri%2Bz-5591Wmt%3DyQTRmrCR8cm5VUQ%40mail.gmail.com.
