Most true statements independent of PA that I know of is equivalent to some consistency statement. For example
- Con(PA), Con(PA + Con(PA)), Con(PA + Con(PA) + Con (PA + Con(PA)), $\dots$
- Goodstein's theorem is equivalent to Con(PA)
- Any conjunction or disjunction of the above.
Is every true statement independent of PA equivalent to some consistency statement?
By "equivalent to some consistency statement", I mean that $PA \vdash S \iff Con(T)$, for some theory $T$. Also, $T$ should be either finite, or specified by a Turing machine that outputs its axioms (and such that PA proves that the Turing machine never stops outputting statements), so that the description of $T$ doesn't throw PA off.
EDIT: In particular, are there are $\Pi^0_1$ examples?