albrecht
Posts:
1,136
Registered:
12/13/04


Re: Can L(<) be the language of the naturals?
Posted:
Sep 2, 2013 2:19 AM


Am Montag, 2. September 2013 05:11:09 UTC+2 schrieb Virgil: > In article <l0023u$jol$1@news.albasani.net>, > > Peter Percival <peterxpercival@hotmail.com> wrote: > > > > > Virgil wrote: > > > > In article <kvvu1c$b1j$2@news.albasani.net>, > > > > Peter Percival <peterxpercival@hotmail.com> wrote: > > > > > > > >> David Hartley wrote: > > > >>> In message <52236CD3.1030800@osu.edu>, Jim Burns <burns.87@osu.edu> writes > > > >>>> If I say that I have a set with a semiinfinite, > > > >>>> discrete, linear order, (N, <), is that enough to > > > >>>> define the naturals? > > > >>> > > > >>> I'm afraid not. Thee are many other orderings satisfying your axioms. > > > >>> E.g. N + Z  i.e. a copy of N followed by a copy of Z. > > > >> > > > >> Also, there is no recursive set of first order axioms that will capture > > > >> just the natural numbers. > > > > > > > > What's wrong with the von Neumann model? > > > > > > What I meant was, there is no first order theory T such that all models > > > of T are isomorphic to the (von Neumann, if you wish) natural numbers. > > > The upward LöwenheimSkolem theorem tells us so. > > > > > > If you want categoricity, you will need a second (at least) order theory. > > > > AS far as I can see, {} and x > x\/{x} captures JUST the natural > > numbers and nothing else, and any other basis captures more. > > 
Your so called "natural numbers" are not the natural numbers of 99% of men. Why calling it "natural numbers" in spite of that fact?
The natural numbers of normal people starts with an object or entity or sign and increases in succesive adding further objects or entities or signs step by step.
E.g.:
I II III IIII IIIII IIIIII ...

