2009/6/3 Torgny Tholerus <torgny.domain.name.hidden>:
>
> Quentin Anciaux skrev:
>> 2009/6/3 Torgny Tholerus <torgny.domain.name.hidden>:
>>
>>> 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?
You just did.
You shown that by assuming there is one it entails a contradiction.
> Have you examined all
> the natural numbers?
No, that's what demonstration is all about.
> How do you prove that there is no biggest number?
You did it.
>
> >
>
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Received on Wed Jun 03 2009 - 13:31:56 PDT