Godel's incompleteness theorems
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From: zuhair on 30 Mar 2010 13:01 Suppose we have a set theory T, which cannot define the notion of "natural number", would that make T escape the incompleteness theorems of Godel's. Zuhair
From: MoeBlee on 30 Mar 2010 13:16 On Mar 30, 12:01 pm, zuhair <zaljo...(a)gmail.com> wrote: > Suppose we have a set theory T, which cannot define > the notion of "natural number", would that make T > escape the incompleteness theorems of Godel's. It's not a matter of defining 'natural number'. Rather, it's a matter of being able to do a certain amount of arithmetic in the theory. We don't have to define 'natural number' just to do arithmetic. First order PA doesn't define, in the theory, 'natural number', but first order PA is rich enough to do enough arithmetic so that it is incomplete. So the fact that a theory doesn't define 'natural number' doesn't in itself entail that the theory is complete. MoeBlee
From: zuhair on 30 Mar 2010 13:42 On Mar 30, 12:16 pm, MoeBlee <jazzm...(a)hotmail.com> wrote: > On Mar 30, 12:01 pm, zuhair <zaljo...(a)gmail.com> wrote: > > > Suppose we have a set theory T, which cannot define > > the notion of "natural number", would that make T > > escape the incompleteness theorems of Godel's. > > It's not a matter of defining 'natural number'. Rather, it's a matter > of being able to do a certain amount of arithmetic in the theory. We > don't have to define 'natural number' just to do arithmetic. First > order PA doesn't define, in the theory, 'natural number', but first > order PA is rich enough to do enough arithmetic so that it is > incomplete. > > So the fact that a theory doesn't define 'natural number' doesn't in > itself entail that the theory is complete. > > MoeBlee What is exactly meant by "enough" arithmetic? is there a definition of that "enough"? Zuhair
From: Daryl McCullough on 30 Mar 2010 13:55 zuhair says... > >Suppose we have a set theory T, which cannot define >the notion of "natural number", would that make T >escape the incompleteness theorems of Godel's. If T does not have a notion of natural number, then it is possible for T to be complete. For example, the theory of real closed fields is decidable and complete. http://en.wikipedia.org/wiki/Real_closed_field#Decidability_and_quantifier_elimination -- Daryl McCullough Ithaca, NY
From: Frederick Williams on 30 Mar 2010 14:23
Daryl McCullough wrote: > > zuhair says... > > > >Suppose we have a set theory T, which cannot define > >the notion of "natural number", would that make T > >escape the incompleteness theorems of Godel's. > > If T does not have a notion of natural number, then it is possible for T to be > complete. For example, the theory of real closed fields is decidable and > complete. What do you mean by having a notion of natural number? In any field one has 0, 1, 1 + 1, 1 + 1 + 1, ... . > > http://en.wikipedia.org/wiki/Real_closed_field#Decidability_and_quantifier_elimination -- I can't go on, I'll go on. |