2009/6/3 Torgny Tholerus <tor...@dsv.su.se>:
>
> Bruno Marchal skrev:
>> On 02 Jun 2009, at 19:43, Torgny Tholerus wrote:
>>
>>
>>> Bruno Marchal skrev:
>>>
>>>> 4) The set of all natural numbers. This set is hard to define, yet I
>>>> hope you agree we can describe it by the infinite quasi exhaustion by
>>>> {0, 1, 2, 3, ...}.
>>>>
>>>>
>>> Let N be the biggest number in the set {0, 1, 2, 3, ...}.
>>>
>>> Exercise: does the number N+1 belongs to the set of natural numbers,
>>> that is does N+1 belongs to {0, 1, 2, 3, ...}?
>>>
>>
>>
>> Yes. N+1 belongs to {0, 1, 2, 3, ...}.
>> This follows from classical logic and the fact that the proposition "N
>> be the biggest number in the set {0, 1, 2, 3, ...}" is always false.
>> And false implies all propositions.
>>
>
> No, you are wrong.  The answer is No.
>
> Proof:
>
> Define "biggest number" as:
>
> a is the biggest number in the set S if and only if for every element e
> in S you have e < a or e = a.
>
> Now assume that N+1 belongs to the set of natural numbers.
>
> Then you have N+1 < N or N+1 = N.
>
> But this is a contradiction.  So the assumption must be false.  So we
> have proved that N+1 does not belongs to the set of natural numbers.

Hi,

No, what you've demonstrated is that there is no biggest number (you
falsified the hypothesis which is there exists a biggest number). You
did a "demonstration par l'absurde" (in french, don't know how it is
called in english). And you have shown a contradiction, which implies
that your assumption is wrong (there exists a biggest number), not
that this number is not in the set.

Regards,
Quentin


> --
> Torgny Tholerus
>
> >
>



-- 
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