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"?

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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...) > 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?) > >> >> It is not possible for "a set" to have no limit. As soon as you >> construct "a set", then that set will always have a limit. >> > > > Is there something intrinsic to your concept of the word "set" that makes > this true? Is your concept of a set fundamentally different than my concept > of well-defined criteria for deciding if any given object is a member or not? > Yes, the definition of the word "all" is intrinsic in the concept of the word "set". -- Torgny --~--~---------~--~----~------------~-------~--~----~ 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 -~----------~----~----~----~------~----~------~--~---