Le Thursday 29 November 2007 18:25:54 Torgny Tholerus, vous avez écrit :
> Quentin Anciaux skrev:
> > Le Thursday 29 November 2007 17:22:59 Torgny Tholerus, vous avez écrit :
> >> There is a difference between "unlimited" and "infinite". "Unlimited"
> >> just says that it has no limit, but everything is still finite. If you
> >> add something to a finite set, then the new set will always be finite.
> >> It is not possible to create an infinite set.
> >
> > I'm sorry I don't get it... The set N as an infinite numbers of elements
> > still every element in the set is finite. Maybe it is an english
> > subtility that I'm not aware of... but in french I don't see a clear
> > difference between "infini" and "illimité".
>
> As soon as you talk about "the set N", then you are making a "closure"
> and making that set finite.
Ok then the set R is also finite ?
> 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.
What is the production rules of the "no"set R ?
> >> So it is OK to use the word "unlimited". But it is not OK to use the
> >> word "infinite". Is this clear?
> >
> > No, I don't see how a set which have not limit get a finite number of
> > elements.
>
> 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.
I don't get it.
> Either you
> have to accept that the set N is finite, or you must stop talking about
> "the set N". It is enough to have a production rule for natural numbers.
I don't accept and/or don't understand.
> >> Another important word is the word "all". You can talk about "all
> >> events". But in that case the number of events will be finite, and you
> >> can then talk about "the last event". But you can't deduce any
> >> contradiction from that, because that is forbidden by the type theory.
> >> And there will be more events after "the last event", because the number
> >> of events is "unlimited".
> >
> > If there are events after the last one, how can the last one be the last
> > ?
>
> The last event is the last event in the set of "all" events. But
> because you have a production rule for the events, it is always possible
> to produce new events after the last event. But these events do not
> belong to the set of "all" events.
There exists no last element in the set N.
> >> As soon as you use the word "all", you will
> >> introduce a limit - all up to this limit. And you must then think of
> >> only doing conclusions that are legal according to type theory.
> >
> > o_O... could you explain what is type theory ?
>
> Type theory is one of the solutions of Russel's paradox. You have a
> hierarchy of "types". Type theory says that the "all quantifiers" only
> can span objects of the same "type" (or lower types). When you create
> new objects, such that "the set of all sets that do not belong to
> themselves", then you will get an object of a higher "type", so that you
> can not say anything about if this set belongs to itself or not. The
> same thing with "the set of all sets". You can not say anything about
> if it belongs to itself.
>
> >> So the best thing is to avoid the word "all" (and all synonyms of that
> >> word).
> >
> > like everything ?
>
> Yes... :-)
What you are saying seems like to me "So the best thing is to avoid words at
all (and any languages)"...
Regards,
Quentin Anciaux
--
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Received on Thu Nov 29 2007 - 13:03:02 PST