Re: An uncountable countable set

In article <45251b3e@xxxxxxxxxxxxxxxxxxx>,
Tony Orlow <tony@xxxxxxxxxxxxx> wrote:

Virgil wrote:
In article <4523c954$1@xxxxxxxxxxxxxxxxxxx>,
Tony Orlow <tony@xxxxxxxxxxxxx> wrote:

David R Tribble wrote:
Tony Orlow wrote:
On the other hand
I don't know why I said "neither can the reals". In any case, the only
way the ordinals manage to be "well ordered" is because they're defined
with predecessor discontinuities at the limit ordinals, including 0.
That doesn't seem "real"
Virgil wrote:
In what sense of "real". There are subsets of the reals which are order
isomorphic to every countable ordinal, including those with limit
ordinals, so until one posits uncountable ordinals there are no
problems.
Tony Orlow wrote:
The real line is a line, with
each point touching two others.
That's a neat trick, considering that between any two points there is
always another point. An infinite number of points between any two,
in fact. So how do you choose two points in the real number line
that "touch"?

They have to be infinitely close, so actually, they have an
infinitesimal segment between them. :)

But any "infinitesimal segment" within the reals is bisectable.

Within the standard reals, it's one number, if it's closer than any
finite distance of a that number.

In Standard reals,"infinitesimal", if it means anything, merely means
very small but not zero.
In The Robinson, or similar, non-standard models, infinitesimals are
different from standard numbers but still non-zero.
In both, they are bisectable, and between two distinct numbers, even
when only infinitesimally different, there is always another.
.

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