Re: Well Ordering the Reals

boink said:
> On Mon, 21 Nov 2005 16:05:42 -0500, Tony Orlow wrote:
>
> > boink said:
> >> On Mon, 21 Nov 2005 15:42:07 -0500, Tony Orlow wrote:
> >>
> >>
> >> > How is this proof different than the inductive proof of the finiteness of the
> >> > values of the natural numbers?
> >>
> >> How does what you just said contradict what I said?
> >>
> >> Any finite set of finite numbers is finite. (Duh.)
> >>
> >> The set of _all_ finite numbers is infinite.
> >>
> >>
> > But, you cannot get an infinite set by starting with 1 root element, and
> > successively adding individual successors, since any finite set will still be
> > finite after the addition of a new element.
>
> However, after _infinitely_ many steps you'll get an infinite set.
>
> > This is the same argument given for
> > the finiteness of the natural numbers, that adding 1 to a finite natural will
> > never give an infinite value, so therefore all successors in the naturals are
> > finite. So, you either accept both proofs and have a finite set of finite
> > naturals, or reject them and have an infinite set with infinite values in it.
> > So, what's it going to be, boinky.
>
> how about both? ...in some sense, that is...
>
> omega = {0, 1, 2, ...} is an infinite set that DOES NOT CONTAIN ANY
> INFINITE elements.
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. You are new to my game, so I'll be
patient - for a bit. :)
>
> omega + 1 contains all finite ordinals AND omega.
Did you just add one element to a finite set and pretend it's infinite? Oh, by
the way, I don't believe in omega. If you can't deal with that, you might want
to drop this hot potato.
>
>

--
Smiles,

Tony
http://www.people.cornell.edu/pages/aeo6/WellOrder/
.

Relevant Pages

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