Are there examples of independence results over subsystems of true second order arithmetic that cannot be established using omega-models? To rule out trivial examples, let us assume that the base theory extends true first order arithemtic. A non example of such a statement would be Ramsey theorem for pairs since there is a computable coloring of pairs of integers into two colors without a computable (even from the halting problem, Jockusch) infinite homogeneous set.


An example of a use of non-omega models appears [here](http://math.berkeley.edu/~slaman/papers/SRT22.pdf) - Also, see the introduction and question 6.1 in this paper.


I am not an expert in reverse mathematics so please feel free to offer any interesting known facts known to you here.



Necessity of omega-models in second order arithmetic
Yes, of course there are. Any result that is equivalent to extra induction axioms will require nonstandard models for a semantic separation.


For example, Hirst proved that the pigeonhole principle $\mathsf{RT}^1\_{<\infty}$ is equivalent to $B\Sigma^0\_2$, and thus is not provable in $\mathsf{RCA}\_0$. But $B\Sigma^0\_2$ holds in every $\omega$-model.


Similarly, there is a variation of $\mathsf{ACA}\_0$, denoted $\mathsf{ACA}'\_0$, which has an axiom asying that for all $n$, $0^{(n)}$ exists. In contrast, $\mathsf{ACA}\_0$ only proves that $0^{(n)}$ exists for standard $n$. So every $\omega$-model of $\mathsf{ACA}\_0$ is a model of $\mathsf{ACA}'\_0$.


For an example with non-arithmetical induction: $\mathsf{ACA}\_0$ plus $\Sigma^1\_1$ induction proves the consistency of $\mathsf{ACA}\_0$, so $\mathsf{ACA}\_0$ does not prove $\Sigma^1\_1$ induction.


