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?