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Topic: Another AC anomaly?
Replies: 280   Last Post: Dec 27, 2009 3:53 PM

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 Dik T. Winter Posts: 7,899 Registered: 12/6/04
Re: Another AC anomaly?
Posted: Dec 2, 2009 10:27 AM
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In article <a7ae8268-3626-4a9f-9902-65662d39d7a3@m25g2000yqc.googlegroups.com> WM <mueckenh@rz.fh-augsburg.de> writes:
> On 2 Dez., 14:14, "Dik T. Winter" <Dik.Win...@cwi.nl> wrote:
...
> > You are conflating within limit cardinality the limit of the cardinalities
> > and the cardinality of the limit. You have to prove they are equal.

>
> If the limit set exists, then it has a cardinal number, hasn't it?

Right, I never denied that.

> > > If it does not exist, set theory claiming the existence of actual
> > > infinity is wrong.

> >
> > Both can exist but they are not necessarily equal, you have to *prove*
> > that.

>
> If a set exists (and if ZFC is correct), then that set has a
> cardinality. If the limit set exists, then it has a cardinality. I
> call that the cardinality of the limit set, abbreviated by limit
> cardinality.

Limit cardinality is confusing because it can either mean the cardinality
of the limit set or the limit of the cardinalities.

> If the limit of cardinalities differs, then either the
> calculation is wrong or the theory whereupon the calculation is based.

Why? I have given you a precise definition of the limit of a sequence of
sets. With that definition the limit of cardinalities is different from
the cardinality of the limit, as is easily calculated. So, what part of
the theory is wrong?

> It is so simple. There is as little proof required as in necessary to
> show that you are Dik T. Winter.

Pray show that I am Dik T. Winter.

> >
> > You have never looked at the definition I think. Given a sequence of sets
> > S_n then:
> > lim sup{n -> oo} S_n contains those elements that occur in infinitely
> > many S_n
> > lim inf{n -> oo} S_n contains those elements that occur in all S_n from
> > a certain S_n (which can be different for each
> > element).
> > lim{n -> oo} S_n exists whenever lim sup and lim inf are equal.
> > With this definition lim{n -> oo} S_n exists and is equal to {}.

>
> Then the theory is wrong.

What part of the theory is wrong?

> If another set exists and the calculation of
> its cardinality gives not the cardinality of that set, what would you
> conclude?

There are not two different sets. What are you babbling about?

> Well, in set the theory the limit set is claimed to exist.

Right. See above for the definition.

> > >http://www.hs-augsburg.de/~mueckenh/GU/GU12.PPT#394,22,Folie22
> >
> > I see nothing related there. It just shows (I think) an open cube and a
> > cylinder.

>
> You must move on. There is always a number remaining in the cylinder
> (the example stretches only until 6, but you should imagine how it
> continues).

I have no idea what you are talking about. I see only a picture of a
cylinder with a few numbers on it and an open cube.

> It shows: If you union all natural numbers within the cube, then it is
> said that you get all natural numbers, the complete set N. But if you
> union all natural numbers such that each one makes an intermediate
> stop in the cylinder and does not move on before the next one, n+1,
> has dropped in, then you cannot union all natural numbers within the
> cube. This simple example shows, that it is impossible to union all
> natural numbers at all. Students easily understand, that this union is
> but a silly idea of unmathematical people.

I do not understand this at all.

> > > > and so
> > > > 2 = lim{n -> oo} | S(n) | != | lim{n -> oo} S_n | = 0
> > > > or can you show what I wrote is wrong?

> > >
> > > Yes I can.* If actual infinity exists*, then the limit set exists.

> >
> > I have still no idea what the mathematical definition of "actual
> > infinity" is, but given the definitions above, the limit of the
> > sequence of sets exists and is the empty set.

>
> Actual infinity is completed infinity.

I have no idea what the mathematical definition of "completed infinity" is.

> Unless actual infinity is assumed, no infinite counting comes to an
> end. No diagonal number comes ever into being.

As I have no idea about what "actual infinity" is, this makes no sense to me.

> > > Then the limit set has a cardinal number which can be determined
> > > simply by counting its elements.
> > > A simple example is
> > > | lim[n --> oo] {1} | = | {1} | = lim[n --> oo] |{1}| = 1.

> >
> > Right. And as in the above example the limit is the empty set, we can
> > count the elements and come at 0.

>
> This proves set theory wrong. In the vase (and in the cylinder in
> http://www.hs-augsburg.de/~mueckenh/GU/GU12.PPT#394,22,Folie 22
> there is always at least one element. Hence cardinality is 1 in the
> limit.

No, that is the limit of the cardinalities.

> > > If you were right, that | lim[n --> oo] S_n | =/= lim[n --> oo] |S_n|
> > > was possible, then the limit set would not actually exist (such that
> > > its elements could be counted and a different limit could be shown
> > > wrong).

> >
> > The limit set is empty.

>
> The limit set is the same as in case 1, 1, 1, ... --> 1
> It is never empty and has never cardinality 0.

No, you are confusing limits of sets and limits of numbers. The
case
{1}, {1}, {1}, {1} --> {1}
is different from
{1}, {2}, {3}, {4} --> ?

> > > > If so, what line is wrong?
> > >
> > > Wrong is your definition of limit set, or set theory claiming actual
> > > infinity as substantially existing, or both.

> >
> > What is *your* definition of the limit of a sequence of sets?

>
> There is no actual infinity. Hence there is no limit set. The only
> thing you can do is this: You can look into my cylinder at any time
> you like and you will find at least one element inside.
>
> Please note the difference: The limit of the *sequence* (1/n) is 0.
> But there is no term 1/n = 0. The limit is not assumed by the
> sequence. It is defined by means of epsilon.

Right, and the limit of a sequence of sets is a set, but that limit is not
assumed by the sequence. It is defined by ohter means (see higher up).

> The limit of ({1, 2, 3, ...n}) however is assumed to exist as N (and
> my cylinder being empty). Our discussion shows that this assumption is
> untenable.

But the limit is not assumed by the sequence. I do not understand why you
do allow the limit of the sequence (1/n) to be 0 but not the limit of the
sequence {n} be {}.
--
dik t. winter, cwi, science park 123, 1098 xg amsterdam, nederland, +31205924131
home: bovenover 215, 1025 jn amsterdam, nederland; http://www.cwi.nl/~dik/

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11/27/09 Dik T. Winter
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