The use of $\aleph$ is quite arbitrary in that context, especially outside set theory.
I have seen $\aleph$ used as a variable for a general (well orderable) cardinal, as a symbol for the continuum, and it does not surprise me that someone whose mathematical background may not include set theory would confuse it for the first transfinite cardinal (i.e. $\aleph_0$).
However indexed $\aleph$ symbols are of course quite well understood as $\aleph_\alpha$ being the $\alpha$-th cardinal.
The $\beth$ symbol is used in mathematics a lot less often, it is the cardinality of the power sets, that is:
- $\beth_0 = \aleph_0$;
- $\beth_{\alpha+1} = 2^{\beth_\alpha}$;
- If $\lambda$ is a limit ordinal, then $\beth_\lambda = \sup\{\beth_\alpha\mid\alpha<\lambda\}$.
I have not seen these used quite often outside set theory, and not often within set theory. As with $\aleph$ symbols, indexed notation has a clear definition.
$\beth_1=|\mathbb R|$, so the continuum hypothesis is $\aleph_1=\beth_1$. If you think on $\aleph$ and $\beth$ as class functions from the ordinals into the cardinals, then the Generalized Continuum Hypothesis asserts that $\aleph=\beth$.
Regardless to all that, coming from the Hebrew side of the screen, $\aleph$ is the first letter of the Hebrew alphabet, and $\beth$ is the second. I do not recall the Hebrew alphabet being ever referred to as "transfinite cardinals" though :-)
(Another interesting addition is that there exists a $\gimel$ (Gimel) function as well in cardinal arithmetics, $\gimel(\kappa)=\kappa^{\operatorname{cf}(\kappa)}$. Gimel is the third letter of the Hebrew alphabet following $\aleph,\beth,\gimel$. Seems that cardinal arithmetic is a lesson in Hebrew, much like mathematics is a lesson in Greek)
(One should be aware that it can be an apparently quite the confusion about the continuum hypothesis, and some mathematicians believe that $\aleph_1$ is defined as $2^{\aleph_0}$)