Why is this function from ordinals to cardinals not used more often?

Question

The Aleph function $f$ is defined from ordinals to cardinals as $f(x)$=the $x$-th infinite cardinal. However, I think a different function $g$ should be used instead. I define $g(x)$= the $x$-th cardinal. So, for example, for any natural number, $g(n)=n$, and $g(\omega)=\omega$, and $g(\omega+1)=\aleph_1$. So, historically, why was the aleph function used more often than this function? This function seems more natural. This is not really a mathematical question, more like a soft question or philosophical question.