User blog comment:Edwin Shade/Ordinal Questions/@comment-1605058-20171128082743

1. \(\frak c\) is commonly used to denote both the cardinality of continuum and the smallest ordinal of its size. This is the notation I would recommend using.

2. To the best of my knowledge there is no standard notation for ordinals representing \(\beth_\alpha\). You can simply write \(\beth_\alpha\) for the ordinals as well, but context should make it clear that you are using ordinal arithmetic (same applies to \(\frak c\)).

3. "The plane can be covered by \(\aleph_1\) lines" is equivalent to CH - to see this, it's clear such a coverimg exists under CH, and for the converse, it's easiest to note that for a fixed circle, every line covers at most two of its points, so we need at least continuum many lines to cover the circle, let alone the whole plane.

It's worth noting that the converse implication is not as immediate as one might think - since lines and planes have the same number of points, the argument "the plane has size continuum times the size of a line, so we need continuum many lines" doesn't work; otherwise, by the same reasoning, we would need at least continuum many planes to cover a plane!

4. I would say the answer is no. In fact, I would argue that the answer is no even for \(n=1\) - while under CH \(\omega_1\) and a unit interval have the same size, they are very different orderings - the former is a well-order, while the latter is a dense ordering. CH doesn't tell us anything about any structure apart from size, and since you explicitly ask about orderings - n-cubes are no good for that.

5. We often refer to it as the "omega fixed point". IIRC it can be expressed using OCFs as \(\psi(\psi_I(0))\), but please correct me if I'm mistaken.