On 31 Jul 2011, at 17:08, Craig Weinberg wrote:

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On Jul 31, 9:49 am, Bruno Marchal <marc...@ulb.ac.be> wrote:In which theory? The notion of proof is theory and definition dependent. (contrary to computability, which is absolute, by Church thesis). If you agree to define x < y by Ez(z+x = y) "E" = "It exists". I assume classical logic + the axioms: x+0 = x x+s(y) = s(x+y) 0 denotes the number zero, and s(x) denotes the successor of x, often noted as x+1. Cf the whole theory I gave last week. I use only a subset of that theory here. So we have to prove that 0 < s(0). By the definition of "<" above, we have to prove that Ez(z + 0 = s(0)) But s(0) + 0 = s(0) by the axiom x + 0 = x given above.So Ez(0 + z = s(0)) is true, with z = s(0). (This is the usual useofthe existence rule of classical logic). Of course we could have taken the theory with the unique axiom "1 isgreater than 0". For all proposition we can always find a theorywhichproves it. The interesting thing consists in proving new fact in some fixed theory, and change only a theory when it fails to prove a fact for which we have compelling evidences.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.`

`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.`

Bruno

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