For a given cardinal number $\aleph_{\alpha}$ we define $$X_{\alpha}= \{\aleph_{\beta}; \aleph_{\beta}<\aleph_{\alpha}\}.$$ We can easily prove that
1) $card(X_{\alpha}) \leq \aleph_{\alpha}^{+}=2^{\aleph_{\alpha}},$ and
2) if $\alpha$ is a successor cardinal number, then $card(X_{\alpha}) \leq \aleph_{\alpha}.$
Now, can we show that $card(X_{\alpha}) \leq \aleph_{\alpha}$ in general? What we can say about the cardinality of $X_{\alpha?}$