I asked this question about two weeks ago on [MSE](https://math.stackexchange.com/questions/1149637/non-forcing-and-independence) 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**?



Non-Forcing and Independence
The [Gödel-Rosser sentence](http://en.wikipedia.org/wiki/Rosser%27s_trick) $R$ 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.


