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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 16, 2009 8:06 AM
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On 16 Dez., 03:15, "Dik T. Winter" <Dik.Win...@cwi.nl> wrote:
> In article <364fe723-5ae4-4ed5-9219-c6b3892b3...@d21g2000yqn.googlegroups.com> WM <mueck...@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.


> Yes, so the definition uses the axiom of infinity.  Without that axiom omega
> would not be immediately existing, as I stated.


No, you stated that you had never seen there a proper definition of
omega.

> 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.


And omega is something that does never end.
>
>  > 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.


No, that's more than that. Omega already is actual infinity. And here
are some other statements about actual infinity:

Edward Nelson:

Let us distinguish between the genetic, in the dictionary sense of
pertaining to origins, and the formal. Numerals (terms containing
only
the unary function symbol S and the constant 0) are genetic; they are
formed by human activity. All of mathematical activity is genetic,
though the subject matter is formal.
Numerals constitute a potential infinity. Given any numeraal, we can
construct a new numeral by prefixing it with S.
Now imagine this potential infinity to be completed. Imagine the
inexhaustible process of constructing numerals somehow to have been
finished, and call the result the set of all numbers, denoted by |N.
Thus |N is thought to be an actual infinity or a completed infinity.
This is curious terminology, since the etymology of ?infinite? is
?not
finished?.
We were warned.
Aristotle: Infinity is always potential, never actual.
Gauss: I protest against the use of infinite magnitude as something
completed, which is never permissible in mathematics.
We ignored the warnings.
With the work of Dedekind, Peano, and Cantor above all, completed
infinity was accepted into mainstream mathematics. Mathematics became
a faith-based initiative.
Try to imagine |N as if it were real. [...]

[Edward Nelson (Department of Mathematics Princeton University):
"Hilbert?s Mistake",
Talk given at the Second New York Graduate Student Logic Conference,
March 18, 2007.]

Samul Feferman:

"I am convinced that the platonism which underlies Cantorian set
theory is utterly unsatisfactory as a philosophy of our subject [...]
platonism is the medieval metaphysics of mathematics; surely we can
do
better" [Solomon Feferman: "Infinity in Mathematics: Is Cantor
Necessary?"]

"The actual infinite is not required for the mathematics of the
physical world"
[Solomon Feferman: IN THE LIGHT OF LOGIC, p. 30]

Regards, WM



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Read Re: Another AC anomaly?
Virgil

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