Second order arithmetic

Godel's theorems are in the context of first order logic. I have a
little knowledge about them, but I have very little knowledge about
second order logic. I have read that the completeness theorem, amongst
other things, fails for second order logic.

More specifically, obviously the formal system of second order
arithmetic can't give us an algorithmic procedure for deciding the
truth of arbitrary arithmetical statements. What kind of arithmetical
statement, then, is unprovable by the means of second order arithmetic?

.

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