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.
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 -- You received this message because you are subscribed to the Google Groups "Everything List" group. To post to this group, send email to everything-list@googlegroups.com. To unsubscribe from this group, send email to everything-list+unsubscr...@googlegroups.com. For more options, visit this group at http://groups.google.com/group/everything-list?hl=en.
