Re: An uncountable countable set

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:
In the sense that the real world is continuous, and you don't just have
these beginnings with nothing before them. 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. :)
.

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