Re: Well Ordering the Reals

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.

omega + 1 contains all finite ordinals AND omega.

.

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