On Sunday, May 19, 2019 at 12:12:13 AM UTC-6, Bruce wrote: > > On Sun, May 19, 2019 at 10:27 AM Russell Standish <li...@hpcoders.com.au > <javascript:>> 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 <mar...@ulb.ac.be > <javascript:>> wrote: > > > > On 16 May 2019, at 03:27, Bruce Kellett <bhkel...@gmail.com > <javascript:>> wrote: > > > > On Thu, May 16, 2019 at 12:59 AM Bruno Marchal <mar...@ulb.ac.be > <javascript:>> > > 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. >
*I find this discussion of Peano's postulates very interesting. FWIW, I usually agree with your views. ISTM that we get our ideas of numbers and arithmetic by viewing the external world. We see many different things out there, so we get the idea of "many". Sometimes we see one of a kind, and can imagine another, and another, leading to the idea of 2 and 3 And we can infer an unending collection of that original one of a kind, leading to the inference of a countable set. Addition is implicit, and multiplication is really addition. So arithmetic realism seems like a huge stretch, to say the least. Now this combination of observation and inference are summarized in Peano's postulates, but with the empiricism eliminated. So they don't seem to accomplish anything, and insofar as empiricism is eliminated, they seem less than meets the eye. So my question is this; why are they important; what do they tell us that we don't already know? AG* > > 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 > ... -- 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/e702030b-7373-4bb9-9620-b9bfbf1d1660%40googlegroups.com.
