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---------------------------------------- > Date: Thu, 29 Nov 2007 19:55:20 +0100 > From: [EMAIL PROTECTED] > To: [EMAIL PROTECTED] > Subject: Re: Theory of Everything based on E8 by Garrett Lisi > > > Jesse Mazer skrev: >> >> >>> From: [EMAIL PROTECTED] >>> >>> >>> 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". Jesse --~--~---------~--~----~------------~-------~--~----~ You received this message because you are subscribed to the Google Groups "Everything List" group. To post to this group, send email to [EMAIL PROTECTED] To unsubscribe from this group, send email to [EMAIL PROTECTED] For more options, visit this group at http://groups.google.com/group/everything-list?hl=en -~----------~----~----~----~------~----~------~--~---