> Date: Fri, 30 Nov 2007 09:00:17 +0100
> Subject: Re: Theory of Everything based on E8 by Garrett Lisi
> Jesse Mazer skrev:
>>> Date: Thu, 29 Nov 2007 19:55:20 +0100
>>> 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.
> What do you mean by a "well-defined criterion"?  Is this a well-defined 
> criterion? :
> The set R is defined by:
> (x belongs to R) if and only if (x does not belong to x).
> If it fits the criterion (x does not belong to x), then it's a member of 
> the set R.
> Then we ask the question: "Is R a member of the set R?".  How shall we 
> use the criterion to answer that question?
> If we substitute R for x in the criterion, we will get:
> (R belongs to R) if and only if (R does not belong to R)...
> What is wrong with this?

My instinct would be to say that a "well-defined" criterion is one that, given 
any mathematical object, will give you a clear answer as to whether the object 
fits the criterion or not. And obviously this one doesn't, because it's 
impossible to decide where R fits it or not! But I'm not sure if this is the 
right answer, since my notion of "well-defined criteria" is just supposed to be 
an alternate way of conceptualizing the notion of a set, and I don't actually 
know why "the set of all sets that are not members of themselves" is not 
considered to be a valid set in ZFC set theory.

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