Quentin Anciaux skrev:
> 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.
>   

How do you know that there is no biggest number?  Have you examined all 
the natural numbers?  How do you prove that there is no biggest number?

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