What things does ZFC not know if it knows? The statement "ZFC $\vdash 0=1$" is independent of ZFC due to Goedel's second incompleteness theorem. That got me wondering, for what other statements $\phi$ is "ZFC $\vdash \phi$" independent of ZFC?
Now of course, for any statement $\phi$ for which "ZFC $\nvdash \phi$" is true, then "ZFC $\vdash \phi$" is independent of ZFC. So instead, I'll ask which "ZFC $\vdash \phi$" are independent of ZFC + Con(ZFC), and stronger theories?
(Of course, easy examples are Con(T) for T=ZFC+Con(ZFC) or stronger theory, but are there others?)
 A: You seem to use "independent of $T$" to mean "unprovable in $T$", so I'll  interpret the question that way (not as "neither provable nor refutable in $T$).
If $ZF\vdash\phi$ is true, then it can be proved in ZF and in fact in much weaker systems, just by taking a proof of $\phi$ in ZF and verifying that it is indeed a proof (and ends with $\phi$). Conversely, if $ZF\vdash\phi$ is false, then it can't be proved in ZF or in ZF + Con(ZF) or in any arithmetically sound theory. (My Platonism is showing here, as I take it for granted that ZF + Con(ZF) is arithmetically sound because it's true.) So $ZF\vdash\phi$ is unprovable in ZF + Con(ZF) iff it is false, i.e., if $\phi$ is unprovable in ZF.
A: I think you're asking what happens if you iterate adding consistency statements.  There's a whole book about that, "Inexhaustibility" by Torkel Franzen, though it starts with arithmetic rather than ZFC.  This article is also good:
https://xorshammer.com/2009/03/23/what-happens-when-you-iterate-godels-theorem/
