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Topic: Can L(<) be the language of the naturals?
Replies: 35   Last Post: Sep 10, 2013 2:12 AM

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 Virgil Posts: 8,833 Registered: 1/6/11
Re: Can L(<) be the language of the naturals?
Posted: Sep 1, 2013 11:11 PM

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 semi-infinite,
> >>>> 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 Löwenheim-Skolem 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.
--

Date Subject Author
9/1/13 Jim Burns
9/1/13 Jim Burns
9/1/13 David Hartley
9/1/13 Peter Percival
9/1/13 Virgil
9/1/13 Peter Percival
9/1/13 Virgil
9/2/13 albrecht
9/6/13 albrecht
9/6/13 Robin Chapman
9/6/13 Tucsondrew@me.com
9/6/13 LudovicoVan
9/6/13 Tucsondrew@me.com
9/7/13 albrecht
9/6/13 Michael F. Stemper
9/7/13 albrecht
9/6/13 FredJeffries@gmail.com
9/7/13 albrecht
9/7/13 FredJeffries@gmail.com
9/8/13 albrecht
9/6/13 Robin Chapman
9/6/13 Brian Q. Hutchings
9/7/13 albrecht
9/6/13 LudovicoVan
9/7/13 albrecht
9/7/13 LudovicoVan
9/8/13 albrecht
9/8/13 LudovicoVan
9/8/13 albrecht
9/9/13 LudovicoVan
9/10/13 albrecht
9/1/13 Jim Burns
9/2/13 Shmuel (Seymour J.) Metz
9/2/13 Shmuel (Seymour J.) Metz
9/2/13 Shmuel (Seymour J.) Metz
9/2/13 Peter Percival