TITLE: Is every true statement independent of $PA$ equivalent to some consistency statement?
QUESTION [7 upvotes]: 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?

REPLY [7 votes]: $1$-consistency of $PA$ is a true $\Pi_3$ sentence which is not provable in $PA$+{all true $\Pi_1$ sentences} (see this article). Simple (iterated) consistency statements (as you mentioned above) are all (true) $\Pi_1$ sentences, so it is not equivalent to any $\Pi_1$ sentence.