A Finite Model of Peano Arithmetic
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From: Frederick Williams on 4 Mar 2010 09:32 RussellE wrote: > Heck, I would be enough to know the FOL statement for primes > with the smallest G�del number. You are seeking a formula phi(x) , the meaning of which is "x is a prime number", and among all the formulae psi(x) such that phi(x) <-> psi(x) you want the one with the smallest G�del number. Is that it? A few points: i) If you can express "x is a prime number" you have gone beyond FOL and are in the realm of arithmetic. ii) Which formula has the smallest G�del number will depend on the numbering scheme you use. iii) If, in answer to ii), you want to use G�del's own numbering, then it certainly isn't FOL you're talking about, its (very roughly) PA + PM. If you want a modest first order theory then Robinson's arithmetic may suit you.
From: Frederick Williams on 4 Mar 2010 09:37 RussellE wrote: > > On Mar 3, 7:19 pm, Arturo Magidin <magi...(a)member.ams.org> wrote: > > On Mar 3, 6:10 pm, RussellE <reaste...(a)gmail.com> wrote: > > > > > Start with Peano's axioms:http://en.wikipedia.org/wiki/Peano_axioms. > > > If an axiomatic theory is consistent, it will still be consistent if > > > we > > > remove an axiom. > > > > > I will remove axiom 7: 0 is not the successor of any natural number. > > > I can now define the set {0,1} and show it is a model of PA. > > > > > Define a successor function: > > > S(0) = 1 > > > S(1) = 0 > > > > As near as I can tell, this set is a model of PA-Axiom 7. > > > > So then, this is not a model for Peano Arithmetic, is it? > > This is a model of Peano arithmetic. It satisfies the 15 axioms > of Peano arithmetic given in the Wiki article. Look again, little looney lad, look again. No. 7 is For every natural number n, S(n) = 0 is False. but in your model S(S(0)) = 0.
From: Aatu Koskensilta on 4 Mar 2010 09:37 Frederick Williams <frederick.williams2(a)tesco.net> writes: > If you want a modest first order theory then Robinson's arithmetic may > suit you. Robinson arithmetic does not come equipped with any G�del numbering. Why should it satisfy Russell's logical needs in any way? -- Aatu Koskensilta (aatu.koskensilta(a)uta.fi) "Wovon man nicht sprechan kann, dar�ber muss man schweigen" - Ludwig Wittgenstein, Tractatus Logico-Philosophicus
From: Frederick Williams on 4 Mar 2010 09:41 Aatu Koskensilta wrote: > > Frederick Williams <frederick.williams2(a)tesco.net> writes: > > > If you want a modest first order theory then Robinson's arithmetic may > > suit you. > > Robinson arithmetic does not come equipped with any G�del numbering. Why > should it satisfy Russell's logical needs in any way? I know, but Russell is taking about FOL and I was suggesting the most modest, that I am aware of, addition to FOL. Nothing "comes equipped with" any G�del numbering, does it? I can only guess at what Russell's logical needs are.
From: Frederick Williams on 4 Mar 2010 09:49
Aatu Koskensilta wrote: > > Frederick Williams <frederick.williams2(a)tesco.net> writes: > > > RussellE wrote: > > > >> Can you express "this statement can't be proven" in FOL? > > > > No because provability, being applicable to formulae, isn't first > > order. > > What do you mean by "first-order" here? Provability is a first-order > concept in a straightforward sense: we can formalize in the first-order > language of arithmetic the notion of provability in any arithmetically > axiomatizable theory. Yes, in first order _arithmetic_, but not in first order _logic_ (which is what I read "FOL" as being). |