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

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mueckenh@rz.fh-augsburg.de

Posts: 18,076
Registered: 1/29/05
Re: Another AC anomaly?
Posted: Dec 2, 2009 9:34 AM
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On 2 Dez., 14:14, "Dik T. Winter" <Dik.Win...@cwi.nl> wrote:
> In article <12b419b5-1a9f-42c1-b9bc-0e8a2cce2...@z41g2000yqz.googlegroups.com> WM <mueck...@rz.fh-augsburg.de> writes:
>  > On 1 Dez., 14:57, "Dik T. Winter" <Dik.Win...@cwi.nl> wrote:
> ...
>  > >  > You may write this as often as you like, but you are wrong. If there
>  > >  > is a limit set then there is a limit cardinality, namely the number of
>  > >  > elements in that limit set. Everything else is nonsense.
>  > >
>  > > Can you prove the assertion that the limit of the cardinalities is the
>  > > cardinality of the limit?
>  >
>  > With pleasure. My assertion is obvious if the limit set actually
>  > exists.
>
> No it is not.
>
>  > Limit cardinality = Cardinality of the limit-set.
>
> 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?
>
>  > 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. If the limit of cardinalities differs, then either the
calculation is wrong or the theory whereupon the calculation is based.
It is so simple. There is as little proof required as in necessary to
show that you are Dik T. Winter. If there is any theory showing that
you are not DTW or that you are 20 meters tall, then that theory is
simply wrong - without further proof.
>
>  > >  I can prove that it can be false.  As I wrote,
>  > > given:
>  > >    S_n = {n, n+1}
>  > > we have (by the definition of limit of sets:
>  > >    lim{n -> oo} S_n = {}
>  >
>  > This is wrong.
>
> 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. If another set exists and the calculation of
its cardinality gives not the cardinality of that set, what would you
conclude?

Well, in set the theory the limit set is claimed to exist.
>
>  >           ong. You may find it helpful to see the approach where I
>  > show a related example to my students.
>  >
>  >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).

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.


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

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.

Regards, WM



Date Subject Author
11/23/09
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11/27/09
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11/27/09
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11/28/09
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11/26/09
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11/30/09
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12/19/09
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12/22/09
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12/7/09
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12/7/09
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12/8/09
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12/27/09
Read Re: Another AC anomaly?
Virgil
12/27/09
Read Re: Another AC anomaly?
Ralf Bader
12/27/09
Read Re: Another AC anomaly?
Virgil
12/13/09
Read Re: Another AC anomaly?
mueckenh@rz.fh-augsburg.de
12/13/09
Read Re: Another AC anomaly?
Virgil
12/4/09
Read Re: Another AC anomaly?
K_h
12/4/09
Read Re: Another AC anomaly?
mueckenh@rz.fh-augsburg.de
12/4/09
Read Re: Another AC anomaly?
Virgil
12/4/09
Read Re: Another AC anomaly?
mueckenh@rz.fh-augsburg.de
12/4/09
Read Re: Another AC anomaly?
Virgil
12/5/09
Read Re: Another AC anomaly?
mueckenh@rz.fh-augsburg.de
12/5/09
Read Re: Another AC anomaly?
Virgil
12/5/09
Read Re: Another AC anomaly?
mueckenh@rz.fh-augsburg.de
12/6/09
Read Re: Another AC anomaly?
Virgil
12/5/09
Read Re: Another AC anomaly?
Carsten Schultz
12/2/09
Read Re: Another AC anomaly?
Virgil
12/1/09
Read Re: Another AC anomaly?
george
12/1/09
Read Re: Another AC anomaly?
Virgil
11/27/09
Read Re: Another AC anomaly?
Virgil
11/27/09
Read Re: Another AC anomaly?
anonymous.rubbertube@yahoo.com
11/30/09
Read Re: Another AC anomaly?
William Hughes
11/27/09
Read Re: Another AC anomaly?
William Hughes
11/27/09
Read Re: Another AC anomaly?
mueckenh@rz.fh-augsburg.de
11/27/09
Read Re: Another AC anomaly?
mueckenh@rz.fh-augsburg.de
11/27/09
Read Re: Another AC anomaly?
William Hughes
11/27/09
Read Re: Another AC anomaly?
anonymous.rubbertube@yahoo.com
11/27/09
Read Re: Another AC anomaly?
mueckenh@rz.fh-augsburg.de
11/27/09
Read Re: Another AC anomaly?
Virgil
11/27/09
Read Re: Another AC anomaly?
mueckenh@rz.fh-augsburg.de
11/27/09
Read Re: Another AC anomaly?
Virgil
11/27/09
Read Re: Another AC anomaly?
mueckenh@rz.fh-augsburg.de
11/27/09
Read Re: Another AC anomaly?
William Hughes
11/27/09
Read Re: Another AC anomaly?
mueckenh@rz.fh-augsburg.de
11/27/09
Read Re: Another AC anomaly?
Virgil

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