LudovicoVan
Posts:
3,201
From:
London
Registered:
2/8/08


Re: Peano's Integers Modulus n.
Posted:
Mar 22, 2014 12:32 PM


"Julio Di Egidio" <julio@diegidio.name> wrote in message news:lgkdrn$mu$1@dontemail.me... > "William Elliot" <marsh@panix.com> wrote in message > news:Pine.NEB.4.64.1403220146430.27476@panix3.panix.com... >> Let N be a set and S:N > N an injection and 0 & z be two elements of N. >> >> Assume Sz = 0 and the schemata, >> if A subset N, 0 in A and for all n in A, Sn in A, >> then A = N. >> >> Are the integers modulus z a model for these axioms? > > Yes, but I don't know how to prove it: how do you prove that something (a > structure?) is a model for some axioms? > >> Is omega_0 + 1 with z = omega_0 another model? > > No, in omega_0 + 1, the successor of omega_0 is omega_0, not zero. > >> Does the inductive definitions: >> for all n,m, 0 + n = n, Sm + n = m + Sn >> 0 * n = 0, Sm * n = m*n + m >> give arithmetic modulus z? > > Addition works, but multiplication does not. For instance: > > 3 * 2 = 2 (mod 4) > 2*2 + 3 = 3 (mod 4)
On the other hand, if you rewrite that as:
0 * n = 0, Sm * n = m*n + n
...
Julio
> Finally, does the model, omega_0 + 1 >> with those axioms yield Peano's arithmetic? > > No: as noticed above, firstly, this is not a model for your axioms, > secondly, in Peano's arithmetic omega_0 isn't a natural number at all. > > Please correct me where I am wrong.

