Let's take a relatively strong axiomatic system, call it $A$. By Godel's Incompleteness Theorem, there are infinitely many true statements expressible in $A$ that cannot be proven. Let's call the set of all truths in $A$ that are provable, $T$. And then all nonprovable truths $NT$.
If we have another axiom system where all the axioms are the negations of $A$'s axioms, i.e $\neg A$, the set of provable theorems is trivially $\neg T$. My question is if the same applies to $NT$, if the set of all unprovable statements in $\neg A$ is $\neg NT$.
In simpler terms, are all unprovable statements in the negation of an axiomatic system the negation of an unprovable statement in the original system?