Why was $\aleph$ (aleph) chosen for infinities?

Solution 1:

Cantor had Jewish roots, which is probably why he was familiar with the Hebrew alphabet. But it's unlikely to be the reason de jure or de facto for the choice.

From Georg Cantor: His Mathematics and Philosophy of the Infinite By Joseph Warren Dauben:

Not wishing to invent a new symbol himself, he chose the aleph, the first letter of the Hebrew alphabet. The choice was especially clever, as he was happy to admit, since the Hebrew aleph served simultaneously to represent the number one, and the transfinite numbers, as cardinal numbers, were themselves infinite unities. (p. 179)

The author then continues with something that looks like an anecdotal ex post facto explanation about how this represented a new beginning.

Solution 2:

According to the book "Set theory, logic and their limitations" by Moshé Machover, aleph is the first letter of the word "einsoph", which is the Hebrew word for infinity, and is also used in cabbalistic traditions as a word for God. Given Cantor's interest in the connection between the infinite and the divine, this seems like the reason for his choice of notation.