# RE: The seven step-Mathematical preliminaries

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> Date: Wed, 3 Jun 2009 13:14:16 +0200
> Subject: Re: The seven step-Mathematical preliminaries
> From: allco...@gmail.com
>
>
> 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.

The English term for this is "proof by contradiction":

Of course, Torgny's conclusion is a little off--he did not show the assumption
"N+1 belongs to the set of natural numbers" must be wrong as he suggested,
rather he showed the assumption "N is the largest natural number" must have
been wrong. Just by the usual definition of natural numbers, if N is a natural
number then N+1 must be one too (the page at
http://en.wikipedia.org/wiki/Recursion#Formal_definitions_of_recursion says
that natural numbers are defined in a recursive way: 'the formal definition of
natural numbers in set theory is: 1 is a natural number, and each natural
number has a successor, which is also a natural number'). If Torgny doesn't
agree, I think he needs to provide an alternate definition of "natural number"
where it would not be true *by definition* that N+1 is a natural number if N is.
Jesse
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