Upon looking at http://plato.stanford.edu/entries/logic-higher-order/, I was a bit confused.
In second order when writing $\forall X$, it seems to mean the power set of the domain of individuals. It seems to me that we use this uppercase variable, because of set of sets issue. But as I read through general semantics parts, I saw that 1-place predicate symbol may or may not be power set of the domain of individuals. I am getting confused. How can this happen? Can't this only equal to $\aleph_0$ if domain of individuals is sized $\aleph_0$ and it is limited to 1-place?