What if a conjecture were provably unprovable?

Question

Suppose we found a proof that "The Twin Prime Conjecture cannot be proven", without any conclusion as to the conjecture itself being true or false.

Is it then possible for the conjecture to be true? or, must the conjecture then be false for some reason? or, must this proof of unprovability be flawed?

(I'm using this specific example to help understand if any similar simple conjectures could be provably unprovable. I have my doubts on whether a simple conjecture can be provably unprovable since, instead of making conclusions for an infinite number of inputs like the Halting Theorem does, we are here talking about just one true/false conclusion.)

Answer 1

There is an important issue with statements of the form "$X$ is unprovable": in what system is $X$ unprovable? We assume here that $X$ is some consistent statement written in some formal language.

Statements provable with no non-logical axioms are known as logical validities. Few interesting mathematical statements are phrased as logical validities. So the fact that $X$ is unprovable without non-logical axioms tells very little about $X$. On the other hand, if $T$ is the conjunction of some finite set of axioms sufficient to prove $X$, then $T \to X$ is a logical validity.

What if $X$ is unprovable from some larger set of non-logical axioms? That mainly tells us that those axioms are not strong enough to prove $X$. In other words, the fact that $X$ is unprovable from a set of axioms often tells us both about $X$ and about that system of axioms. As long as $X$ is consistent, we could always choose a stronger set of consistent axioms that is able to prove $X$ (e.g. we could take $X$ itself as an axiom).

In many concrete examples, the unprovable of $X$ from some system of axioms tells us more about the system of axioms than it tells us about $X$. For example, the combinatorial principle in Goodstein's theorem is unprovable in Peano arithmetic. The main interest in this is that, when we really look at the result, it shows us something new about provability in Peano arithmetic. It also helps us understand the combinatorial principle more deeply, but in this case we already have a very good understanding of the combinatorial principle which comes from its proof in ZFC.

This is a common pattern in mathematical logic. When we examine the proof that a principle is unprovable in a particular system, it often tells us as much or more about the system as it tells us about the principle - particularly when we scrutinize the unprovability proof intensively to try to extract the general method.

Answer 2

If a conjecture X is unprovable that can be translated to the following:

if X is true no contradiction can be derived

if X is false no contradiction can be derived

Both of these must be true because otherwise X would not be unprovable (we could merely choose one of the appropriate routes and eventually find a contradiction)

That being said it becomes evident then that the truth/false value of X is independent of the system of proving being used.

So once something has become unprovable in a fixed system of mathematics. It becomes meaningless to ask if the statement is true or false in that same system.

Hope that clarifies a thing or two :)