On Jul 31, 11:58 am, Bruno Marchal <marc...@ulb.ac.be> wrote:
>
> > How do we know that 0 has a successor though? If 0 x = x and x -0 = x
> > then maybe s(0)=0 or Ez<>s(0)... Can we disprove the idea that a
> > successor to zero does not exist?
>
> No. 0 is primitive term, and the language allows the term s(t) for all
> term t, so you have the terms 0, s(0), s(s(0)), etc.

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It sounds like you're saying that it's a given that 0 has a successor
and therefore doesn't need to be proved.
> The rest follows from the axioms For all x 0 ≠ s(x), s(x) = s(y) -> x
> = y (so that all numbers have only one successor. So you can, prove,
> even without induction, that 0 has a unique successor, different from
> itself.
>
> > Sorry, I'm probably not at the
> > minimum level of competence to understand this.
>
> I look on the net, but I see errors (Wolfram's definition is Dedekind
> Arithmetic!)? On wiki, the definition of Peano arithmetic seems
> correct. You need to study some elementary textbook in mathematical
> logic. Most presentation assumes you know what is first order
> predicate logic. You can google on those terms. There are good books,
> but it is a bit involved subject and ask for some works. Peano
> Arithmetic is the simplest example of Löbian theory or machines or
> belief system. It is very powerful. You light take time to find an
> arithmetical proposition that you can prove to be true and that she
> can't, especially without using the technics for doing that. Most
> interesting theorem in usual (non Logic) mathematics can be prove in
> or by PA. And PA, like all Löbian machine, can prove its own Gödel
> theorem (if "I" am consistent then "I" cannot prove that "I "am
> consistent). The "I" is a 3-I.
Thanks, I'll see if I can nibble on it sometime.
Craig
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