Ordering on the Cardinals - A Question About Set Builder Notation Using Cardinals

Question

I'm currently reading Marker's text on Model Theory, and he constructs a set similar to the following, $$ \{\alpha : \alpha < \kappa \}$$ where $\kappa$ is an infinite cardinal. He claims this set has cardinality at least $\kappa$. I'm a bit confused what this set refers to. I'm not very versed on cardinals, so specifically I'm confused on what the $\alpha$'s are. If the $\alpha$'s are also cardinals, then if we take $\kappa$ to be the continuum $\aleph_1$, wouldn't $\alpha \in \{\aleph_0\} \cup \mathbb N$, since those are all of the cardinals less than $\aleph_1$? In that case, of course the above set doesn't have the same cardinality as the continuum.

So what do the $\alpha$'s refer to?

Answer 1

It should be described when the set is defined. Most likely the $\alpha$ are all the ordinals less than $\kappa$ and it is true that there are $\kappa$ of them. If we take $\kappa$ to be $\aleph_1$ (which may or may not be the continuum) it is the set of all countable ordinals

Answer 2

It is the set of all ordinals $\alpha$ with $\alpha < \kappa$. In fact, this set is equal to $\kappa$ because for ordinals $\in$ = $<$ and every element of an ordinal is again an ordinal.

In other words, as sets we have

$$\kappa=\{\alpha: \alpha < \kappa\}= \{\alpha: \alpha \in \kappa\}$$