I'm trying to see/show that second order logic (with full semantics) is incomplete - i.e. that there are sentences that are true in all models of some theory $T$, and yet still can not be proved from $T$.
First of all, is this even the case? And if it is, is this how it can be shown :
The basic idea is that if second order logic were complete, we should then be able to prove any sentence of arithmetic, which Gödel's theorem says we can not. It seems that Gödel's incompleteness theorem should apply also to second order logic (does it?), and so SOL completeness ($N\vDash \phi$ iff $N\vdash \phi$) would mean that any $\phi$ true in the standard model of number theory (i.e. $N\vDash \phi$ ) is also provable from N, contradicting the incompleteness theorem for SOL.