cardinality of all cardinal numbers less than a cardinal number

Question

For a cardinal number $\alpha$ what is the cardinality of the set

$X=\{\beta, \beta$ is a cardinal number with $\beta<\alpha\}?$

Can we say for example that $card(X)=\alpha$ or $card(X)\leq \alpha?$

Answer 1

The inequality is obvious, because $X\subset \alpha$. The equality, however, is false : for instance if $\alpha=\omega_1$, then $X= \omega+1$, so $Card(X) = \omega$.

I don't think there is a general formula for this; the thing is that if $\kappa$ is a cardinal with $\kappa = \aleph_\alpha$, then $X= \omega\cup\{\aleph_\beta, \beta <\alpha\}$, so that $Card(X) = \aleph_0 + Card(\alpha)$. But $\alpha$ can be $<\kappa$, and it can be $=\kappa$, so there isn't much more you can say