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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 3, 2009 9:46 AM
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On 3 Dez., 13:52, "Dik T. Winter" <Dik.Win...@cwi.nl> wrote:
> In article <0c635f53-f5c4-4320-8825-de05f021a...@m3g2000yqf.googlegroups.com> WM <mueck...@rz.fh-augsburg.de> writes:
>  > On 2 Dez., 16:27, "Dik T. Winter" <Dik.Win...@cwi.nl> wrote:
>  > > Limit cardinality is confusing because it can either mean the cardinality
>  > > of the limit set or the limit of the cardinalities.
>  >
>  > If the limit set is "assumed" by the sequence of sets, if for instance
>  > N is given by unioning all naturla numbers, then tghe limit set has to
>  > have the limit cardinality.
>
> The limit set is not necessarily "assumed" by the sequence of sets (if by
> that you mean that there is a set in the sequence that is equal to the
> limit set, if you mean something else I do not understand it at all).  When
> you mean with your statement about N:
>      N = union{n is natural} {n}
> then that is not a limit.  Check the definitions about it.


It is a limit. That is independent from any definition.
>
> So below is your argument:
>
>  > If you want to prove that the determinant of a matrix M with not
>  > ecclusively linear independent rows is det(M) = 0, you go two ways:
>  > You multiply a row of the matrix M by 0, and you empty a row of M by
>  > elementary operations which do not change the determinant. Then you
>  > have proved that the original matrix M has the determinant 0.
>
> So you argue that each matrix has determinant 0?  If not what do you
> mean with that?
>
>  > Same holds for cardinals of sets. You form a set by unioning its
>  > elements, resulting in the complete set.
>
> You can not unite the elements if the elements are not sets in themselves
> and when the elements are sets, the union of the elements is different
> from the original set.  Consider:
>    A = {{a, b}, {b, c}, {c, d}}
> uniting the elements gives:
>    B = {a, b, c, d}
> quite different.


Not at all different with respect to being a limit process.
>
>  >                                          This cannot change the
>  > connection between cardinal number and set during the whole process of
>  > formation. For every step the cardinal number and the number of
>  > elements of the set are equal.
>
> What do you mean with "every step"?  Uniting sets is a single step
> operation.
>


No. Uniting two sets or singlets or elements is a single-step
operation.
Uniting infinitely many sets or singlets or elements is a limit
process.

>  > Same holds for the cylinder. Its contents is
>  >   {1}, {1}, {1}, {1} , ...
>  > and that is not different from
>  >   {1}, {2}, {3}, {4} , ...
>  > as you can see by renaming the elements.
>
> I do not understand this at all.


I see. You seem to share that fate with most so-called matematicians.
>
>  > I think that this answers all the questions contained in the
>  > following. Therefore I extinguish it.
>
> No.  I asked you for a mathematical definition of "actual infinity" and you
> told me that it was "completed infinity".  Next I asked you for a mathematical
> definition of "completed infinity" but you have not given an answer.  So I
> still do not know what either "actual infinity" or "completed infinity" are.


Both are nonsense. But both are asumed to make sense in set theory.
The axiom of infinity is adefinition of actual infinity.
"There *exists* a set such that ..."
Without that axiom there is only potential infinity, namely Peano
arithmetic.

Regards, WM

Regards, WM


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11/27/09
Read Re: Another AC anomaly?
Virgil

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