Torgny Tholerus wrote:
> Jesse Mazer skrev:
>
>>
>>> Date: Fri, 5 Jun 2009 08:33:47 +0200
>>> From: tor...@dsv.su.se
>>> To: everything-list@googlegroups.com
>>> Subject: Re: The seven step-Mathematical preliminaries
>>>
>>>
>>> Brian Tenneson skrev:
>>>
>>>> How can BIGGEST+1 be a natural number but not belong to the set of all
>>>> natural numbers?
>>>>
>>> One way to represent natural number as sets is:
>>>
>>> 0 = {}
>>> 1 = {0} = {{}}
>>> 2 = {0, 1} = 1 union {1} = {{}, {{}}}
>>> 3 = {0, 1, 2} = 2 union {2} = ...
>>> . . .
>>> n+1 = {0, 1, 2, ..., n} = n union {n}
>>> . . .
>>>
>>> Here you can then define that a is less then b if and only if a belongs
>>> to b.
>>>
>>> With this notation you get the set N of all natural numbers as {0,
>>>
>> 1, 2,
>>
>>> ...}. But the remarkable thing is that N is exactly the same as
>>> BIGGEST+1. BIGGEST+1 is a set with the same structure as all the other
>>> natural numbers, so it is then a natural number. But BIGGEST+1 is not a
>>> member of N, the set of all natural numbers.
>>>
>> Here you're just contradicting yourself. If you say BIGGEST+1 "is then
>> a natural number", that just proves that the set N was not in fact the
>> set "of all natural numbers". The alternative would be to say
>> BIGGEST+1 is *not* a natural number, but then you need to provide a
>> definition of "natural number" that would explain why this is the case.
>>
>
> It depends upon how you define "natural number". If you define it by: n
> is a natural number if and only if n belongs to N, the set of all
> natural numbers, then of course BIGGEST+1 is *not* a natural number. In
> that case you have to call BIGGEST+1 something else, maybe "unnatural
> number".
>
>
I wonder if anyone has tried work with a theory of finite numbers: where
BIGGEST+1=BIGGEST or BIGGEST+1=-BIGGEST as in some computers?

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Brent
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