Le Friday 16 November 2007 09:33:38 Torgny Tholerus, vous avez écrit :
>  Quentin Anciaux skrev:
> Hi,
>
> Le Thursday 15 November 2007 14:45:24 Torgny Tholerus, vous avez écrit :
>
>   What do you mean by "each" in the sentence "for each natural number"? 
> How do you define ALL natural numbers?
>
>
>
> There is a natural number 0.
> Every natural number a has a natural number successor, denoted by S(a).
>
>
>  What do you mean by "Every" here?  Can you give a *non-circular*
> definition of this word?  Such that: "By every natural number I mean
> {1,2,3}" or "By every naturla number I mean every number between 1 and
> 1000000".  (This last definition is non-circular because here you can
> replace "every number" by explicit counting.)

I do not see circularity here... every means every, it means all natural 
numbers possess this properties ie (having a successor), that means by 
induction that N does contains an infinite number of elements, if it wasn't 
the case that would mean that there exists a natural number which doesn't 
have a successor... well as we have put explicitly the successor rule to 
defined N I can't see how to change that without changing the axioms.

>
>
> How do you prove that each x in N has a corresponding number 2*x in E?
> If m is the biggest number in N,
>
>
> By definition there exists no biggest number unless you add an axiom saying
> there is one but the newly defined set is not N.
>
>
>  I can prove by induction that there exists a biggest number:
>
>  A) In the set {m} with one element, there exists a biggest number, this is
> the number m. B) If you have a set M of numbers, and that set have a
> biggest number m, and you add a number m2 to this set, then this new set M2
> will have a biggest number, either m if m is bigger than m2, or m2 if m2 is
> bigger than m. C) The induction axiom then says that every set of numbers
> have a biggest number.
>
>  Q.E.D.
>
>  --
>  Torgny Tholerus

Hmm I don't understand... This could only work on finite set of elements. I 
don't see this as a proof that N is finite (because it *can't* be by 
*definition*).

Quentin Anciaux

--
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