Conjectures which can't be right or wrong Recently I was talking with some of my non-mathematician friends. On some very unrelated subject in order to make my point I said:


  "There are some conjectures in mathematics which are proven to be
  neither right or wrong."


They were really impressed, I was happy, but I don't know if I am right or wrong (I may conject that). Are there such statements and if there are can you tell me please?
What do you mean by neither right nor wrong? 

Certainly for many important specific axiom systems $T$, such as first-order Peano arithmetic, or ZFC (by far the most widely used axiomatic set theory) there are sentences that are neither provable nor refutable from $T$. 

But for example any sentence $\varphi$ in the first-order language of number theory is either true or false in the natural numbers. We just don't happen to know, for many such $\varphi$, whether $\varphi$ is true or false in $\mathbb{N}$. And first-order Peano arithmetic, though very powerful, is provably not strong enough to decide the question for all $\varphi$. Indeed no recursively axiomatized theory is strong enough to determine the answer for all $\varphi$. However, that has no direct connection with truth in $\mathbb{N}$.
The proposition (p$\rightarrow$q), where it is not known whether "p" is true or false, and it is not known where "q" is true or false, is neither right nor wrong, nor could it get proven either way in classical (and plenty of other types of) logic.