TITLE: Clarification of a remark of J. Steel on the independence of Goldbach from ZFC
QUESTION [11 upvotes]: 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.

REPLY [5 votes]: Let me add a footnote to Andreas Blass's answer.
Suppose $T$ is a theory which extends PA (or indeed, extends Robinson Arithmetic). Then, for the reasons indicated, if $T$ is consistent, it is $\Pi_1$ sound. [For suppose $\varphi$ is a false $\Pi_1$ sentence. Then $\neg\varphi$ is a true $\Sigma_1$ sentence and so provable in PA. Hence if $T$ proved $\varphi$ it would be inconsistent. Contrapose.]
So suppose you come up with e.g. some Weird and Wonderful set theory $W\&W$ that can do a modicum of arithmetic. And suppose that in $W\&W$ you can prove e.g. the Riemann Hypothesis or Fermat's Last Theorem. Then you don't have to believe in the existence of $W\&W$'s superduperweirdo cardinals, or whatever. You just have to believe that $W\&W$ is consistent -- and bingo, you have an argument for the Riemann Hypothesis. (Echoes here, of course, of the Hilbert Program!)
I remember thinking this was rather magical when I first came across this observation when the world was very much younger: and students can still find it very striking.