> Date: Fri, 30 Nov 2007 09:00:17 +0100
> From: [EMAIL PROTECTED]
> To: [EMAIL PROTECTED]
> Subject: Re: Theory of Everything based on E8 by Garrett Lisi
> Jesse Mazer skrev:
>>> Date: Thu, 29 Nov 2007 19:55:20 +0100
>>> From: [EMAIL PROTECTED]
>>> 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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