On page 424 of the following paper:
S. Feferman, Harvey M. Friedman, P. Maddy and John R. Steel, ``Does Mathematics Need New Axioms?'' The Bulletin of Symbolic Logic, Vol. 6, No. 4 (Dec., 2000), pp. 401-446
John Steel makes the following remark in footnote 29:
There is the very remote possibility that one could show ZFC settles the questions without actually exhibiting the relevant ZFC-proofs. Goldbach's conjecture and the Riemann hypothesis are $\Pi_1^0$ statements, so one cannot prove them independent of ZFC without also proving them.
Could someone explain to me what kind of result/principle he is referring to here?
I'm guessing it's something of the form: ''For any $\Pi_1^0$ statement $\phi$ in PA (or, possibly, some weaker arithmetic) Ind(ZFC, $\phi$) iff some condition.'' The condition obviously can't be PA $\vdash \phi$ because, by soundness, that would imply that $\phi$ is not independent. But I can't think what other condition would justify Steel's remark.