I am Formalist. And from a formal point of view, GCH is independent from ZF(C) - nothing more. To me, sets are not really "existing" (have you ever seen an infinite ordinal flying around somewhere? well, i didnt.) - but assuming GCH could make mathematics easier - and that is what is important to me. Without GCH, there must be at least one set (and therefore infinitely many sets) with an undecidable cardinality.
On the other hand, with GCH, if you can prove $|A| > \aleph_i$ and $|A| \le \aleph_{i+1}$ then you are already done proving $|A| = \aleph_{i+1}$.
