Re: Well Ordering the Reals


Tony Orlow wrote:

<snip>

>... In any set of
> successive naturals starting at 1, any largest element always equals the set
> size, and any set size always equals the largest element.

The first part is correct (but if you have been doing your execises you
will remember that not all sets have a largest element)

The second part is not correct. While it is true that for sets with
a largest element the set size always equals the largest element, it
it not true that for sets without a largest element that the set size
equals the largest element. We can conclude that the set size
is not a standard natural number and not a TO-natural number,
but we cannot conclude that the set size does not exist.


[You proved that if a largest element exists it must be the set
size. The statement, "if a set size exists it must be the largest
element" does not follow.]

> This is a constant
> equality for every case proven inductively, and there is no reason to imagine
> this breaking down at oo. So, why do you have a set of successive naturals
> starting at 1, with only finite values and no specific largest one, but with a
> supposedly specific set size, which is infinite?

Since aleph null is not a natural number or a TO-natural number
there is no problem.

-William Hughes

.