Truly Untrue?

This post will not compile without Truth Is Buildable. One import, and you can read the rest. A claim is untrue when no build has passed, when some proof or test would settle it and none has closed. Untrue is not false. False is a build that ran and broke. Untrue is the build that never returned. Hold that one definition and follow it somewhere the last essay did not go, up a ladder that does not want to be climbed.

The fifty-year wait

Take a claim that has waited half a century. P equals NP. No one has proved it. No one has refuted it. It is untrue, in the exact sense imported above, and it has been untrue since before most living computer scientists were born. People have a feeling about it. They believe it comes out one way. A feeling is a credence, not a build, and the ledger does not take credence. So it waits.

Most of what you know sat in this state once. Every theorem was a conjecture first. The wait is not failure. It is the working condition of inquiry, the room every truth passes through on the way in. At any long vigil you ask the same question. Will it end? And the harder one underneath. Can I even know whether it will end?

Ask the next question up

Here is the move that should make you uneasy. P equals NP is untrue. Now ask: is it provable from our axioms? Will the wait ever close?

We do not know. No one has shown P equals NP is provable, and no one has shown it independent of our axioms. That is a question with no proof and no refutation, which makes “P equals NP is provable” untrue by the same definition. That question has not closed either. The untruth did not stay on the ground floor. It climbed. You reached for the question that would tell you how long to wait, and that question is also a vigil.

So is the question above that, and the one above it. The untruth climbs rung by rung, and at each rung you say the same thing. That’s untrue. That’s untrue too.

This looks like a trap with no floor. It is not. The floor exists, and finding it is the whole essay.

The hung build

Make the build literal and the shape appears. Searching for a proof is running a program: enumerate candidate proofs, halt when one checks. Then truth is not a mood. It is a halting state.

A proof-search that halts with a proof makes the claim true, the build closed green. One that halts with a refutation makes it false, the build closed red. One that has not halted leaves it untrue, the program still running.

That is the whole trichotomy in one machine, and it tells you something the last essay only asserted. True and false are siblings because they are both halting states, computations that returned a verdict. Untrue is the one state that is not a verdict. It is the build still spinning, and you have seen it a hundred times. The test suite that never finishes. The deploy stuck at ninety percent. The job you leave running overnight and find still running in the morning. Green, red, or hung. You named the first two true and false. Untrue is the hung build.

And now the vigil has a precise question. Watching a hung build, can you tell whether it will finish or hang forever?

Three ways to wait

No. Not in general. That is one of the most famous theorems in the field, and it has been sitting in the undergraduate halting chapter the whole time, wearing a different name.

But “not in general” is not “never.” Sometimes you can tell, and the cases sort into three, which is the trichotomy again, one floor up. Ask of any untrue claim: is it resolvable?

It can be true. Some claims live in a system where a decision procedure is guaranteed. A statement of Presburger arithmetic is decidable, so you know, before you run anything, that the search halts. The wait is real but guaranteed to end. Knowably temporary.

It can be false. Some searches can be shown never to halt, no proof coming and no refutation either. The claim is then out of reach for good, and you can prove it is. Knowably forever.

It can be untrue. P equals NP. You do not know if it is resolvable. The question of how long to wait is itself a hung build.

So the meta-question wears the same three faces as the claim underneath it. Apply “is this resolvable” to an untruth and you get back true, untrue, or false. The verdict that climbed the ladder does not climb forever. At some rungs it lands.

Undecidability is decided

This is the floor. Say it slowly, because it sounds like a paradox and is not.

The halting problem has no solution. But whether the halting problem has a solution does have one. The answer is no, and it is proved. Undecidability, here, is not an open question. It is a closed one. We did not fail to decide it. We decided it, and what we decided is that no general decision exists. The theorem does not settle any single program. It settles whether a method exists to settle them all, and the answer is no.

So the third state is not uniform fog. It has structure, and the first cut in that structure is whether your own wait is knowable. Most untruths leave you in suspense about the suspense. A few come stamped. The halting problem is the proof that “untrue, and provably forever” is a real place a claim can sit, a vigil you are told, with certainty, has no morning. That telling is the strangest mercy in the subject. You do not get the verdict you waited for. You get a different one, that the wait was the verdict, and now you can stop.

The refrain you climbed with changes on this rung. All the way up you said: that’s untrue, that’s untrue too. Here, for once, the rung answers back. That’s decided.

What the machine cannot hand you

It would be clean to stop there and say epistemology is just the halting chapter in a coat. The proof ports one to one. Turing’s diagonal is Turing’s diagonal. But the meaning does not port, and the gap is the reason to write this down.

Computability is silent about the world. It tells you a search does not halt. It does not tell you there is an answer out there the search is failing to reach. The machine is agnostic. The realist is not. P equals NP has a settled answer in real, a definite fact about what algorithms can do, whether or not any proof ever finds it. Undecidable in the ledger. Determinate in the world. The chapter gives you the first sentence. You supply the second, and the second is where untrue stops being a fact about machines and becomes a fact about knowing.

Which sets the one rule for inheriting the rest of the field. You do not get the theorems free because you said the word “build.” You get each theorem you can rebuild on this side, where you check that its premises survive the move from computation to knowledge. For formal claims like proof or decidability, once the system is fixed, the transfer is exact. For an empirical question whose settlement waits on an instrument or a telescope or a grant, not on a Turing machine, the transfer is a metaphor and has to say so. An imported theorem whose premises do not hold here is a claim with no passing build. It is untrue. The field that studies untruth had better not smuggle one in.

Truly untrue?

So, the title. When a claim has waited a long time and you are tired of waiting, you want to know if it is truly untrue, settled in its suspension, never coming, or only untrue for now, a morning away. The honest answer is a third thing. Sometimes you can know. The halting problem is the proof that “forever” is sayable, that a wait can be certified hopeless and closed. Sometimes you cannot, and the question of which kind of wait you are in is itself a wait. P equals NP is untrue. Whether it is truly untrue, untrue all the way down with no proof coming, is also untrue.

The vigil is not blind. It is not always lit either. You hold the claim, you run the build, and you watch for one of three things. The green that says true. The red that says false. Or the rarer message, harder won than either, that says stop watching, this one runs forever, and we can prove it. That last one is not the verdict you wanted. It is still a verdict. In a field where most things never answer, being told for certain that something never will is the closest the machine comes to grace.