> Date: Sat, 6 Jun 2009 16:48:21 +0200
> From: tor...@dsv.su.se
> To: everything-list@googlegroups.com
> Subject: Re: The seven step-Mathematical preliminaries
> 
> 
> 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".

OK, but then you need to define what you mean by "N, the set of all natural 
numbers". Specifically you need to say what number is "BIGGEST". Is it 
arbitrary? Can I set BIGGEST = 3, for example? Or do you have some 
philosophical ideas related to what BIGGEST is, like the number of particles in 
the universe or the largest number any human can conceptualize?
Also, any comment on my point about there being an infinite number of possible 
propositions about even a finite set, or about my question about whether you 
have any philosophical/logical argument for saying all sets must be finite, as 
opposed to it just being a sort of aesthetic preference on your part? Do you 
think there is anything illogical or incoherent about defining a set in terms 
of a rule that takes any input and decides whether it's a member of the set or 
not, such that there may be no upper limit on the number of possible inputs 
that the rule would define as being members? (such as would be the case for the 
rule 'n is a natural number if n=1 or if n is equal to some other natural 
number+1')
Jesse
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