An uncountable countable set [Math]

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From: mueckenh on 6 Oct 2006 05:34

Dik T. Winter schrieb:

> In article <1160079987.659148.278260(a)b28g2000cwb.googlegroups.com> mueckenh(a)rz.fh-augsburg.de writes:
> > Dik T. Winter schrieb:
> ...
> > > > 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."
>
> Yes, that change is still current in set theory, and I know the difference.
> And I even do understand why he used his original notation. What is more
> natural than asserting that with a.b you combine a copies of set b? But
> there were more compelling reasons to interchange the operands.
>
> > 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.
>
> But that notation suggests that there is an ordinal (-k), which is true
> in most branches of algebra.
>
> But whatever. Subtraction and division is nearly nowhere defined or used
> in set theory. There appears to be not much need. So you should not be
> surprised that I question what you write, when that was possibly common
> 120 years ago, but was not taught at all some 40 years ago, when I had my
> set theory courses.
>
> > > > 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.
>
> But in the standard ways algebra works, first it is asserted that the
> negative of a number does exist, and after that subtraction is defined
> as the addition of the negative...
>
> > > 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.
>
> It is not unrestricted possible. There are cases where subtraction is
> not possible at all. Possibly one of the reasons that it is not part of
> mainstream mathematics.
>
> > But why should I stick to an
> > old-fashioned and misleading notation?
>
> No reason at all, but when you invent a new notation, it would be better
> if you did define the notation before use.

Ok, but as we have agreement now, we can return to he main question:
Why do you think that 0.111... with the index sequences 1,2,3,... or
k+1,k+2,k+3 or -k, -k+1, -k+2, ... represents exactly *one* number
only, as you asserted?

Regards, WM

From: Randy Poe on 6 Oct 2006 09:10

mueckenh(a)rz.fh-augsburg.de wrote:
> Dik T. Winter schrieb:
>
> > In article <1160079778.871756.325200(a)m73g2000cwd.googlegroups.com> mueckenh(a)rz.fh-augsburg.de writes:
> > >
> > > Dik T. Winter schrieb:
> > >
> > > > > The axiom of infinity does only state n+1 exists if n is given. It is
> > > > > realized by he numbers of my list.
> > > >
> > > > That is *not* what the axiom of infinity states. The axiom states that
> > > > there exists a set that contains all the successors of its elements.
> > >
> > > Oh I must have read always wrong texts. I never came across the word
> > > "all" in connection with this axiom.
> >
> > What texts *have* you read where your version is stated?
>
> I read many texts. (Don't intermingle the quantity "all" with the
> quanifyer "forall".)

That didn't answer the question. In which one did your
version of the axiom of infinity occur?

- Randy

From: Randy Poe on 6 Oct 2006 09:11

From: Randy Poe on 6 Oct 2006 09:15

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