IJ footnote

[Adam Lou Stephanides]

On Sat, 23 Mar 1996, Andrew Dinn wrote:

> Jeffrey Reid writes:
> 
> > > On Thu, 21 Mar 1996, Adam Lou Stephanides wrote:
> > > > There is at least one other math error in the footnotes: Cantor's
> > > > Diagonal Proof doesn't mean that between any two objects you can put
> > > > an infinity of other objects.
> 
> > It does if those objects are numbers on the real line...
> 
> Sorry, but I have to disagree with both of you.

[stuff deleted]
 
> However, the way I read the quote attributed DFW he claims Cantor's
> proof shows that no infinite collection is complete, in the sense that
> that no matter what infinite cardinality you care to identify there is
> always some infinite cardinality which is infinitely larger. This

I'll have to stand by my original statement.  In the footnote, Cantor is 
described as "the man who proved some infinities were bigger than other 
infinities, and whose 1905-ish Diagonal Proof demonstrated that there can 
be an infinity of things between any two things no matter how close 
together the two things are." (p. 994)  This is almost exactly what I
said, and I don't see how it can have the interpretation you give it.

> And note that Cantor's proof does rely on taking an *ordering* of some
> collection with the first cardinality and showing that there are
> ommissions in the ordering. The jump to assume that all the missing
> elements are `between' the ones in the ordering (which Cantor does
> make) is actually rather a large one (look at it real closely, because
> it's the one that gets you from the rationals to the reals and that's
> where the rot creeps in).

I'm a little confused here.  How does Cantor's Diagonal Proof assume
that all the missing elements are "between" the ones in the ordering?
I don't see him making this assumption in his 1891 statement of the
Diagonal Proof.

--Adam