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

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 Virgil Posts: 870 Registered: 7/27/09
Re: Another AC anomaly?
Posted: Dec 2, 2009 3:47 PM

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

> On 2 Dez., 14:14, "Dik T. Winter" <Dik.Win...@cwi.nl> wrote:
> > In article
> > <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.

To demonstrate that the cardinality of a limit process of a seqeunce of
non-cardinals equals a different sort of limit process of a sequence of
cardinals requires more proof that WM is capable of producing.

> It is so simple.

WM is what is 'simple' here. Too simple to see that something is beyond
his ken.

> 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.

Definitions cannot be 'wrong'. They can be useless, but not wrong.

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

That the two sets are distinct.
>
> Well, in set the theory the limit set is claimed to exist.

Not always. According to the definition above, there is a necessary
condition for that limit, which need not always be met, in which case
the limit does NOT exist.

Thus one may imagine a situation in which the limit of the cardinalities
does exist but the limit of the sequence of sets does not exist.
e.g., a sequence of sets all of the same cardinality but for which
neither lim sup nor lim inf 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.

Unfortunately for WM, the very people whom he calls unmathemtical are
the very ones who create all the mathematics that he pretends to
understand.
>
>

> >
> >  > > 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.

And, by WM's definition, Completed infinity is actual infinity.
Unfortunately, neither is relevant to set theory.

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

Actual one-at-at-a-time counting may not end in situations in which
bijections are known to exist. In which case one-at-a-time counting is
irrelevant to 'counting' of the whole.
> >
> >  > 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.

It only proves WM 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.

That is like saying that the intersection of {1,2,3} and {4,5,6} has 3
elements in it because each set heas three elements in it.
> >
> >  > 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.

WM only gets to command what happens when safe in Wolkenmuekenheim.

Outside of his little private country WM no longer rules.

> 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.

One more than is in his head.
>
> 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.

But it is not a limit of a sequence of sets nor of sequence of
cardinalites, so it is doubly irrelevant.
WM fails by imagining that all limits must behave identically even when
they cannot behave identically.
>
> 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.

WM's discussin may seem to show that, but his discussion is not valid
outside Wolkenmuekenheim.
>
> Regards, WM

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