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Topic: Matheology § 095
Replies: 25   Last Post: Aug 1, 2012 6:25 PM

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 Virgil Posts: 4,674 Registered: 1/6/11
Re: Matheology � 095
Posted: Jul 25, 2012 2:07 PM

In article
WM <mueckenh@rz.fh-augsburg.de> wrote:

> On 24 Jul., 21:03, Virgil <vir...@ligriv.com> wrote:
> > In article

>
> >
> > If what WM means is the sequence of sets {0},{1,2}, {3,4,5,6}, ...,
> > then there is a limit, the empty set, at least according to standard set
> > definitions, but if WM means anything else, like, for example, ordered
> > sets, (0), (1,3), (3,4,5,6), ..., then he must prove his claim using
> > some accepted definition of a limit of appropriate type.

>
> Consider this again:

> > > ((1, 1)) = S_1
> > > ((2, 1), (3, 2)) = S_2
> > > ((4, 1), (5, 2), (6, 3)) = S_3
> > > ...

>

Is LimInf S_n = LimSup S_n ?

If not, then no limit exists!

> If the first coordinates have a limit and the second coordinates have
> a limit, then the set of pairs must have a limit too, at least in case
> set theory consistently describes infinite sets.

Wrong! The only relevant definition for a limit to a
sequence of sets, S_n, is the common value, if there is one,
of LimInf S_n and LimSup S_n.

And since S_n /\ S_m = {} whenever m =/= n, that common value is {}, if
it exists at all.
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