Re: An uncountable countable set

mueckenh@xxxxxxxxxxxxxxxxx wrote:
*** T. Winter schrieb:

In article <1160045362.894290.321140@xxxxxxxxxxxxxxxxxxxxxxxxxxxx> mueckenh@xxxxxxxxxxxxxxxxx writes:
>
> *** T. Winter schrieb:
>
> > In article <1159978513.826507.125470@xxxxxxxxxxxxxxxxxxxxxxxxxxxx> muecke=
> nh@xxxxxxxxxxxxxxxxx writes:
> > > *** T. Winter schrieb:
> > ...
> > > > > You can read it above. There is the order preserving union of a s=
> et of
> > > > > k negative numbers and a set of omega natural numbers including z=
> ero.
> > > >
> > > > But that has ordinal k + omega,
> > >
> > > that is what I said!
> >
> > No.
>
> I used "k + omega " as the ordinal number of {-k, -k+1, ..., 0,
> 1,2,3,...}.

Sorry, I misread.

> > > > not -k + omega. A set of k negative numbers
> > > > has ordinal k; not -k.
> > >
> > > But subtraction of a set of positive numbers from the set omega is
> > > expressed, as I did, by -k + omega.
> >
> > Ah, apparently you are defining something new here.
>
> New for you probably. As omega + k is different from k + omega, we
> should not write omea - k for -k + omega.

Cantor did. And he wrote omega_(-k) for the other.

Cantor later changed notation. Perhaps you don't know that Cantor
changed notation between 1883 and 1895. In 1883 Cantor had not yet done
it. Compare the remark [3] by Zermelo on page 208 of his collected
works: "Hier und im folgenden stellt Cantor den Multiplikator voran und
schreibt 2 omega für omega + omega; in der späteren systematischen
Darstellung III 9 stellt er umgekehrt den Multiplikandus voran und
schreibt omega * 2, was aus Gründen der Analogie entschieden
vorzuziehen ist, weil auch bei der Addition nur der zweite Summand (der
Addendus), wenn er endlich ist, die transfinite Summe modifiziert,
vergrößert. Vgl. S. 302, 322."

In order to use this analogy which Zermelo mentiones I prefer -k +
omega, because so every nutcake sees that the sum is not modified but
remains omega while omega - k could be misunderstood as modifying the
sum. So I do in addition what, according to Zermelo, has to be
preferred in multiplication because of the analogy to addition and
which Cantor executed in 1895 or somewhat earlier.

> > > No? Who decides that? You see, I knew already that, according to
> > > Cantor, subtraction is possible. If I express this as addition of
> > > negative, what do you think did I meant?
> >
> > I did not know because there is no definition presented nor available.
> > Addition is defined between ordinal numbers. Ordinal numbers are (by
> > their very definition) larger than or equal to 0. But you want to
> > define addition between ordinal numbers and non-ordinal numbers.
>
> Wrong again. k and omega are ordinal numbers.

But '-k' is *not* an ordinal number.

"k" is an ordinal number and "-" is the advice to subtract it.

In the meantime I have read it. You may note that in his notation
the solution for x + a = b is b - a (a < b). And the solution for
a + x = b (if that exists, and if there is more than one solution,
the smallest one) as b_(-a).
--

So you have learned that subtraction is possible, which was the main
point that you originally doubted. But why should I stick to an
old-fashioned and misleading notation?

I'm coming into this part of the conversation late, so I hope I have
the context correct.

We should be clear. Of course we can make various definitions, but you
can see that certain of them are conditional definitions. If the
conditions fail, then you cannot apply the definition as if the
condition holds. In this case, we don't have a definition of
subtraction that allows for substracting a finite ordinal k from w
(read 'w' as 'omega'). For a finite ordinal k, we don't have w-k as
some kind of ordinal such that there is an x such that k+x=w or even as
any kind of object (except by some method of assigning a default value
for otherwise non-referring terms, which is a whole other subject).

MoeBlee

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