Non-Forcing and Independence I asked this question about two weeks ago on MSE and haven't gotten an answer, so I thought I would post the question here.
Do there exists sentences which are independent of ZFC, cannot be shown to be independent through some method of forcing, and do not increase the consistency strength of ZFC (e.g. so Large Cardinal Axioms are out)? 
If there does exist such a sentence, I would love to know a concrete example. One with a combinatorial flavor would be ideal. 
Edit: Feel free to increase the consistency strength of ZFC (say ZFC+ $\Delta$) in the "meta-"sense (i.e. work where you want to work). Does there exist a "non-forcible" independent sentence that does not increase the consistency strength of ZFC?
 A: If we consider $ZF$ instead of $ZFC$, then we can say more.
There are examples of such results which are obtained by Krivine, using his method of realizability. For example in the paper Realizability algebras II: New models of ZF+DC the following is stated:

Using the proof-program (Curry-Howard) correspondence, we give a new method to obtain models of $ZF$ and relative consistency results in set theory. We show the relative consistency of $ZF + DC$ + there exists a sequence of subsets of $\mathbb{R}$ the cardinals of which are strictly decreasing + other similar properties of $\mathbb{R}$. 

As it is stated in the introduction of the paper: 

These results seem not to have been previously obtained by forcing.

see also 50 years after forcing, the Curry-Howard correspondence gives new models of ZF
A: The Gödel-Rosser sentence $R$ for $\text{ZFC}$ is an arithmetic assertion, such that $\text{ZFC}$ is equiconsistent with $\text{ZFC}+R$ and with $\text{ZFC}+\neg R$. So the Rosser sentence does not increase consistency strength. Since arithmetic assertions are preserved by forcing, one cannot use forcing directly to prove the independence of $R$. 
Another example would be $\neg\text{Con}(ZFC)$, since $\text{ZFC}$ is equiconsistent with $\text{ZFC}+\neg\text{Con}(\text{ZFC})$, and so this doesn't increase consistency strength. (the theory $\text{ZFC}+\text{Con}(\text{ZFC})$, in contrast, does have strictly higher consistency strength). So this is an arithmetic assertion that is independent of $\text{ZFC}$, assuming that $\text{ZFC}$ is consistent, but this is not possible to prove in any direct way by forcing, since forcing does not affect arithmetic truth.
