Conjectures which can't be right or wrong

Question

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?

Answer 1

Any mathematical theory depends on a set of axioms. Once the axioms are fixed, you can discuss the existence of undecidable propositions. You can find a list here.

Answer 2

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 $\mathbb{N}$.

Answer 3

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.

Answer 4

You mention in particular conjectures which are proven 'neither right or wrong'. As others have mentioned in answers, perhaps the best way to go about making the idea of 'right or wrong' precise is in terms of undecidability. It is a consequence of Gödel's incompleteness theorems that for any reasonably strong axiomatic theory, there are statements which are neither provable nor refutable using that theory.

For example, Gödel's first incompleteness theorem shows that we can construct a sentence which roughly says

This sentence is not provable in theory $T$

which isn't provable in $T$ (assuming that $T$ is consistent). (For if it were provable, it would be true, and hence not provable. Contradiction.)

The second incompleteness theorem says that the sentence:

Theory $T$ is consistent

is not provable in $T$ (again, assuming that $T$ is consistent).

But these perhaps aren't best suited to what you're getting at because (and note that these are more philosophical reasons than mathematical):

1. These perhaps aren't best described as mathematical conjectures. They have a particular metamathematical character, and, in some cases (especially in the first incompleteness theorem) a rather ad hoc character—constructed specifically for that purpose.

2. There is a very strong intuition that, although these sentences are undecidable in the theory, they are nonetheless either right or wrong, and the incompleteness theorems simply show up the limits of the theory. For example, the first sentence above, which says that it is not provable, we have proved not to be provable. Hence, it's true!. Similarly, it seems that there's a fact of the matter whether some theory is consistent or not.

But there are some sentences which don't seem to share these characteristics (again, depending on certain philosophical views). The most famous example is Cantor's Continuum Hypothesis. Cantor had proven that the set of all real numbers is larger in size than the set of all natural numbers. But then he conjectured that it's, in some rigorous sense, the next size up; there are no sets which are intermediate in size between the natural numbers and the real numbers. This claim is the Continuum Hypothesis. It was a genuine conjecture, which Cantor (and I think others) put quite some effort into proving or refuting.

This sentence is independent of the most commonly accepted axioms for set theory—the ZFC axioms (proved partly by Gödel, and partly by Cohen). Moreover, it has proved to be the case that most natural extensions to ZFC also fail to decide the continuum hypothesis. There aren't really any easy ways to see whether it is true or false, as there are in the case of the first two sentences. Whether this is a good reason to think that it is 'neither right nor wrong' is more a matter of philosophy than mathematics, but opinion amongst set theorists seems to be fairly divided on the matter. It certainly seems like a better candidate than those involved in the incompleteness theorems.