Sum question? Sum from $\infty$ to $\aleph_0$

Question

$\sum^{\aleph_0}_{\mu=\infty}\mu=\aleph_0$? If this is true, however, then $\sum^{\aleph_0}_{\mu=0}\mu=\aleph_1$?

Answer 1

As far as I'm aware, the expressions $\sum_{\mu=\infty}^{\aleph_0} \mu$ and $\sum_{\mu=0}^{\aleph_0} \mu$ don't have definitions. If an expression hasn't been defined, then it doesn't have a value.

In general, there's no definition of a summation that has $\aleph_0$ as one of its bounds.