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...)
> 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
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