Le 16-nov.-07, à 09:33, Torgny Tholerus a écrit :

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>>>>
>> 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.)
>
>>
>>> 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.
What do you mean by "every" here?
You just give us a non ultrafinitistic proof that all numbers are
finite, not that the set of all finite number is finite.
Bruno
>
> Q.E.D.
>
> --
> Torgny Tholerus
>
> >
>
http://iridia.ulb.ac.be/~marchal/
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