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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 15, 2009 9:15 PM
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In article <364fe723-5ae4-4ed5-9219-c6b3892b3df2@d21g2000yqn.googlegroups.com> WM <mueckenh@rz.fh-augsburg.de> writes:
> On 11 Dez., 03:40, "Dik T. Winter" <Dik.Win...@cwi.nl> wrote:
...
> > > > Without the axiom of infinity omega would not be immediately
> > > > existinging.
> > > > So apparently there is a definition of omega without the axiom of
> > > > infinity.
> > > > Can you state that definition?

> > >
> > > Look into Cantor's papers. Look into my book.

> >
> > I have never seen there a proper definition of omega.
> >

> I knew you did not read it.

I did read it.

> Look at p. 93. There the natural numbers are constructed.
> On p. 86 all ordinals till eps^eps^eps^... are given. On p. 90 you see
> the axiom of infinity 7.


Yes, so the definition uses the axiom of infinity. Without that axiom omega
would not be immediately existing, as I stated. But you thought there was
a definition without that axiom. I still have not found such a definition.
(The definition there is:
omega = {0, 1, 2, 3, ...}
and without the axiom of infinity it is not clear whether the right hand side
is a set.)

> > > > There are no concepts of mathematics without definitions.
> > >
> > > So? What is a set?

> >
> > Something that satisfies the axioms of ZF for instance.

>
> Is that a definition?


That is not something unheard of. In mathematics a ring is something that
satisfies the ring axioms, and that is pretty standard.

> But in case you shouldn't have been able to find a definition of
> actual infinity, here is more than that: omega + 1.


Ok, so actual infinity now is omega + 1. In other articles you stated that
it was a theory ("according to actual infinity") and various other things.

> > > If an infinite set exists as a limit, then it has gotten from the
> > > finite to the infinite one by one element. During this process there
> > > is no chance for any divergence between this set-function and its
> > > cardinality.

> >
> > If a function exists as a limit, then it has gotten from the finite to
> > the infinite one by one step. During this process there is no chance
> > of any divergence between the function and the integral.
> >
> > Now, what is wrong with that reasoning?
> >
> > Stronger:
> > lim(n -> oo) 1/n = 0
> > 1/n > 0

>
> I do not understand. Do you see a gap here?


Getting from something > 0 to something equal to 0.

> > If a number exists as a limit, then it has gotten from the finite to the
> > infinite one by one step. During this process there is no chance of any
> > divergence between the element and the inequality.
> >
> > What is wrong with that reasoning?
> >
> > You are assuming that taking a limit is a final step in a sequence of
> > steps.
> > In the definition I gave for the limit of a sequence of sets there is no
> > final step.

>
> I did not say that there is a final step. I say that there is no
> chance for a difference of lim card(S_n) and card(lim(S_n)) where lim
> means n --> oo.


Yes, you just say that. Without proof.

> If, in your example you would claim that lim(1/n) = 0 and and 1/omega
> = 10, I would not accept such behaviour as mathematics (like your
> funny Sum n = 0 or Euler's -1/12).


What I state is:
1/n > 0
and
lim(n -> oo) 1/n = 0
and you state that there is no chance for a difference of
1/n > 0 for all n and lim(n -> oo) 1/n = 0

About that "funny" sum, that "funny" sum has no definition, so it is allowed
to give it a definition. But apparently you do not allow that.

Yes, I know you dislike definitions and axioms, because they constrain you.
--
dik t. winter, cwi, science park 123, 1098 xg amsterdam, nederland, +31205924131
home: bovenover 215, 1025 jn amsterdam, nederland; http://www.cwi.nl/~dik/



Date Subject Author
11/23/09
Read Re: Another AC anomaly?
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of Bentley College
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12/2/09
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of Bentley College
Marshall
11/27/09
Read Re: Another AC anomaly?
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11/27/09
Read Re: Another AC anomaly?
William Hughes
11/26/09
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mueckenh@rz.fh-augsburg.de
11/28/09
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mueckenh@rz.fh-augsburg.de
11/28/09
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Virgil
11/26/09
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William Hughes
11/28/09
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William Hughes
11/26/09
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mueckenh@rz.fh-augsburg.de
11/28/09
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11/28/09
Read Re: Another AC anomaly?
Virgil
11/29/09
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mueckenh@rz.fh-augsburg.de
11/29/09
Read Re: Another AC anomaly?
Virgil
11/29/09
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ross.finlayson@gmail.com
11/29/09
Read Re: Another AC anomaly?
Marshall
11/30/09
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Virgil
11/30/09
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ross.finlayson@gmail.com
11/30/09
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ross.finlayson@gmail.com
11/30/09
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11/30/09
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11/30/09
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11/29/09
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11/26/09
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11/26/09
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11/26/09
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11/29/09
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11/27/09
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11/27/09
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11/30/09
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11/30/09
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11/27/09
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Read Re: Another AC anomaly?
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11/27/09
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11/27/09
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11/27/09
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11/27/09
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11/27/09
Read Re: Another AC anomaly?
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11/27/09
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mueckenh@rz.fh-augsburg.de
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11/30/09
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mueckenh@rz.fh-augsburg.de
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Read Re: Another AC anomaly?
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12/1/09
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mueckenh@rz.fh-augsburg.de
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12/19/09
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12/22/09
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12/22/09
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12/18/09
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12/19/09
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12/21/09
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12/21/09
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12/27/09
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12/16/09
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12/16/09
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12/11/09
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12/11/09
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12/11/09
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12/12/09
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12/10/09
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12/7/09
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12/3/09
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12/3/09
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12/3/09
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12/7/09
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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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12/8/09
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12/10/09
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12/16/09
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12/16/09
Read Re: Another AC anomaly?
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12/17/09
Read Re: Another AC anomaly?
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12/4/09
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mueckenh@rz.fh-augsburg.de
12/4/09
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12/4/09
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12/7/09
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12/7/09
Read Re: Another AC anomaly?
mueckenh@rz.fh-augsburg.de
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Read Re: Another AC anomaly?
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12/8/09
Read Re: Another AC anomaly?
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12/10/09
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12/11/09
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12/4/09
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12/5/09
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11/27/09
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11/27/09
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11/27/09
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11/27/09
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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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