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.
--
Torgny Tholerus
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