# n-place relation function in general/standard semantics in second-order language

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?

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This question doesn't make much sense to me. Can you please clarify, everything? – Asaf Karagila Jul 15 '12 at 6:07