The Friedman-Goldfarb-Harrington (FGH) principle for an arithmetical theory T states that for any Sigma_1-sentence S there is a (Sigma_1) sentence R such that "Provable[R]" is T-equivalent to "S or Provable[0=1]". In particular, any Sigma_1 sentence S that follows from inconsistency is equivalent to "Provable[R]" for a suitable R.

For T=S₂¹ we deal with sentences of the form (exists)(Sigma_1^b), because such is the complexity of the provability predicate, and ask whether the corresponding statement holds.

The problem arises in S₂¹, because the standard proof uses a Rosser-type fixed point construction of R:

R <-> "S is witnessed before Provable[R]".

Such an R is (exists)(Pi_1^b) and one then would want to apply the (exists)(Pi_1^b)-completeness principle, which is not available.

A related argument was used by Friedman to prove the equivalence of the Sigma_1-disjuction property and the Sigma_1-reflection property (modulo consistency). This results in a similar kind of problem in bounded arithmetic.