How to show incompleteness of second order logic?

Question

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.

Answer 1

The problem is that just saying "any proof system" doesn't give you enough to work with. There is certainly some complete proof systems -- for example the one where every logically valid second-order formula is declared to be an axiom. On the other hand that proof system is not effective, which makes it not very exciting.

However, you're right that you can use Gödel's incompleteness theorem to show that there is no proof system that is all of {sound, complete, effective}. We can even take $T$ to be the empty theory.

The key to this is that the Peano axioms can be expressed as a finite conjunction of formulae in pure second-order logic, so having a sound proof system for second-order logic will allow us to prove formulas $\varphi$ about integer arithmetic by asking whether $$ \forall 0\,\forall S\,\forall {+}\,\forall{\times}\,\Bigl[\text{Peano}(0,S,{+},{\times}) \to \varphi \Bigr] $$ is provable in the empty theory. (This is standardly true in all universes if and only if $\varphi$ is true in the integers).

Either there are easy arithmetic truths that the proof system fails to prove in this way (in which case it is clearly incomplete), or it proves enough that Gödel's theorem applies to it.