Godel's incompleteness theorems
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From: Nam Nguyen on 30 Mar 2010 23:42 zuhair 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. What do we mean by "escape the incompleteness theorems"? That is, what is the sense of "escape" here? Any rate, there's a system T with it's L(T) = L(0,*,<), and with infinite number of prime numbers. I've been wondering if such T would be incomplete but GIT would have no say about that, but haven't got a clear idea.
From: zuhair on 31 Mar 2010 07:20 On Mar 30, 10:42 pm, Nam Nguyen <namducngu...(a)shaw.ca> wrote: > zuhair 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. > > What do we mean by "escape the incompleteness theorems"? > That is, what is the sense of "escape" here? > > Any rate, there's a system T with it's L(T) = L(0,*,<), and > with infinite number of prime numbers. I've been wondering > if such T would be incomplete but GIT would have no say about > that, but haven't got a clear idea. I meant escape the criterion of "sufficiency for arithmetic" that is present in the clause of GIT. MoeBlee somewhat answered this question. Zuhair
From: Don Stockbauer on 31 Mar 2010 09:25 On Mar 31, 6:20 am, zuhair <zaljo...(a)gmail.com> wrote: > On Mar 30, 10:42 pm, Nam Nguyen <namducngu...(a)shaw.ca> wrote: > > > zuhair 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. > > > What do we mean by "escape the incompleteness theorems"? > > That is, what is the sense of "escape" here? > > > Any rate, there's a system T with it's L(T) = L(0,*,<), and > > with infinite number of prime numbers. I've been wondering > > if such T would be incomplete but GIT would have no say about > > that, but haven't got a clear idea. > > I meant escape the criterion of "sufficiency for arithmetic" that > is present in the clause of GIT. MoeBlee somewhat answered this > question. > > Zuhair Before you do anything, you need to jump to the metalevel and decide if it's worth doing.
From: John Jones on 1 Apr 2010 06:13 zuhair 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. > > Zuhair Goedels theorem is based on the idea that a framework can be found that is big enough to embrace different objects, and not just objects with different values. These objects can be anything whatever. So if there is a set theory T which doesn't employ natural numbers, then a framework can be found that is big enough to make it an element in a framework that does employ natural numbers.
From: Jesse F. Hughes on 1 Apr 2010 08:08
John Jones <jonescardiff(a)btinternet.com> writes: > zuhair 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. >> >> Zuhair > > Goedels theorem is based on the idea that a framework can be found that > is big enough to embrace different objects, and not just objects with > different values. These objects can be anything whatever. You *really* shouldn't answer honest mathematical questions with your ignorant pseudo-philosophical bluff. It's not nice. What if Zuhair honestly tries to understand what you wrote? It would be a pointless waste of his time. Keep your bullshit to your own threads. > So if there is a set theory T which doesn't employ natural numbers, then > a framework can be found that is big enough to make it an element in a > framework that does employ natural numbers. -- "Clouds are always white and the sky is always blue, And houses it doesn't matter what color they are, And ours is made of brick." -- A new song by Quincy P. Hughes (age 4) |