Re: Well Ordering the Reals

In article <1133427464.857888.284340@xxxxxxxxxxxxxxxxxxxxxxxxxxxx>,
albstorz@xxxxxx wrote:

> David Kastrup wrote:
> > albstorz@xxxxxx writes:
> >
> > > David Kastrup wrote:
> > >> albstorz@xxxxxx writes:
> > >>
> > >> > David Kastrup wrote:
> > >> >> boink <boink@xxxxxxxxxx> writes:
> > >> >>
> > >> >> > On Mon, 21 Nov 2005 16:34:20 -0500, Tony Orlow wrote:
> > >> >>
> > >> >> >> How did the set become infinite, if you only added a finite number
> > >> >> >> of elements to it? if you added elements an infinite number of
> > >> >> >> times, each one bigger than the last, how do you NOT have elements
> > >> >> >> which are the result of an infinite number of successions, and if
> > >> >> >> you do, how do they NOT have infinite values? If, asfter infintiely
> > >> >> >> many steps you get an infinite set, then after infinitely many
> > >> >> >> increments, you get an infinite value.
> > >> >> >
> > >> >> > that's funny, because in some sense that's exactly what happens, but
> > >> >> > you don't get it. after omega many steps, you get the infinite
> > >> >> > ordinal omega which is the set of all finite ordinals. and omega is
> > >> >> > infinite and contains no infinite ordinal.
> > >> >>
> > >> >> Nonsense. With that kind of logic, with aleph_1 many steps, you
> > >> >> get the infinite ordinal omega_1, but there is no such thing.
> > >> >> Steps don't get you omega. Omega is an _inexhaustible_ supply of
> > >> >> sequential steps. Either you have it, or you don't. If you don't,
> > >> >> you can't put it together using finite steps. A sentence like "you
> > >> >> get omega, if you just exhaust an inexhaustible supply of finite
> > >> >> steps..." does not make sense.
> > >> >
> > >> >
> > >> > It's really funny. You have infinitely many natural numbers but you
> > >> > have not infinitely many steps (or you have it but you can't supply
> > >> > the natural numbers with it).
> > >>
> > >> You have an inexhaustible supply of steps. Since it is inexhaustible,
> > >> you can't perform "all of those".
> > >
> > > This must hold in the exact same manner for the inexhaustible supply of
> > > natural numbers. Since it's inexhaustible you can't have all of
> > > those.
> >
> > Not one by one. Only as a supply. And N is exactly that.
>
> If N is a supply, its extent is exact this what is usually called
> potential infinity.
Then we only have a potential one and a potential two , etc., as these
have no more actuality than N does.

> > This is not problematic: _all_ mathematical entities exist by virtue
> > of their definition and their properties.
> >
>
> That's arrogance. The natural numbers don't exist since you or someone
> else define them. The natural numbers exist, since men exists. 100.000
> years ago, nobody had defined N.
> Not math creates N. N creates math!

Mankind creates both!

Or does Storz wish to reserve that right of creativity to God?
>
> Regards
>
> Albrecht Storz
.

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