Specifically, I'm talking about the set $\displaystyle S_{\Omega}$, the minimal uncountable well-ordered set, every section of which is countable.

I'm curious about two things.

The first is: Are there any (recognizable) representations for this set? It's driving me crazy to come up with an example for the thing.

The second is: What is the cardinality of this set? It has to be greater than the cardinality of the integers (being that it's uncountable), but I can't see that it can be as big as the cardinality of the real numbers (being that every section of the reals is uncountable) and, by the Continuum Hypothesis, there is no cardinal number between the two.

Thanks.

Dan