I think it is well-known that $PA+\neg con(PA)$ is $\Pi\_1$-conservative over $PA$ (for proof see Smorynski's article, 'the incompleteness theorems', in handbook of mathematical logic).


What can we say about $PA+\neg R\_{PA}$ ? is it also $\Pi\_1$-conservative over $PA$ ?($R\_{PA}$ is the Rosser sentence for $PA$).



Is $PA+ \neg R_{PA}$ $\Pi_1$- conservative over $PA$?
The Rosser sentence $R$ asserts, "for every proof of $R$ in PA,
there is a smaller proof of $\neg R$." So $\neg R$ asserts, "there
is a proof of $R$ in PA, with no smaller proof of $\neg R$."


In particular, we may deduce in $\text{PA}+\neg R$ that "for every
proof of $\neg R$, there is a smaller proof of $R$". This is a
$\Pi^0\_1$ assertion.


But this implication is not provable in PA alone, since $R$ is
known (under suitable consistency assumptions) to be strictly
weaker than $\text{Con}(\text{PA})$, and in any model of
$\text{PA}+R+\neg\text{Con}(\text{PA})$ there will be proofs of
both $R$ and $\neg R$, but because $R$ is true, the smallest proof
of $\neg R$ will be strictly smaller than the smallest proof of
$R$.


