Alon: "Saharon Shelah, in particular, published extensively on the question of what Aleph could the cardinality of the continuum possibly be."
Actually this question (and much more) is answered by a theorem of W.B. Easton from 1970. It turns out that the continuum could be any infinite cardinality of uncountable cofinality! (At least if we assume that the axioms ZFC are consistent, which most people are happy to do.)
See Easton's Theorem, which is proved by a variation of the forcing method originally used by Paul Cohen to get the consistency of that ZFC + not-CH is equiconsistent with ZFC:
So, e.g. the continuum could be aleph_1, or aleph_17, or even the 23rd Woodin cardinal (if you believe in such things), but it could NOT be aleph_omega (this would violate Konig's Lemma).
Shelah has made many contributions to "cardinal arithmetic" and what kinds of bounds can be proved for cardinal exponents just in ZFC. This gets pretty technical pretty fast, but you can get the flavor of it here: