> Date: Thu, 29 Nov 2007 19:55:20 +0100
> Subject: Re: Theory of Everything based on E8 by Garrett Lisi
> Jesse Mazer skrev:
>>> As soon as you talk about "the set N", then you are making a "closure" 
>>> and making that set finite.
>> Why is that? How do you define the word "set"? 
>>   The only possible way to talk about 
>>> something without limit, such as natural numbers, is to give a 
>>> "production rule", so that you can produce as many of that type of 
>>> objects as you want.  If you have a natural number n, then you can 
>>> "produce" a new number n+1, that is the successor of n.
>> Why can't I say "the set of all numbers which can be generated by that 
>> production ruler"?
> As soon as you say "the set of ALL numbers", then you are forced to 
> define the word ALL here.  And for every definition, you are forced to 
> introduce a "limit".  It is not possible to define the word ALL without 
> introducing a limit.  (Or making an illegal circular definition...)

Why can't you say "If it can be generated by the production rule/fits the 
criterion, then it's a member of the set"? I haven't used the word "all" there, 
and I don't see any circularity either.

>>  It almost makes sense to say a set is *nothing more* than a criterion for 
>> deciding whether something is a member of not, although you would need to 
>> refine this definition to deal with problems like Russell's "set of all sets 
>> that are not members of themselves" (which could be translated as the 
>> criterion, 'any criterion which does not match its own criterion'--I suppose 
>> the problem is that this criterion is not sufficiently well-defined to 
>> decide whether it matches its own criterion or not).
> A "well-defined criterion" is the same as what I call a "production 
> rule".  So you can use that, as long as the criterion is well-defined.
> (What does the criterion, that decides if an object n is a natural 
> number, look like?)

I would just define the criterion recursively by saying "1 is a natural number, 
and given a natural number n, n+1 is also a natural number".

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