What Is Truth? Your Smoke Alarm Can Say "Yes." It Can Never Say "Never."

Your smoke alarm has been silent for 1,000 days.

Does that prove your house will never have smoke in it?

No. Day 1,001 can bring the beep. A thousand silent days leave "never" exactly where it started: unproved.

Now watch what one beep does. The moment the alarm goes off, "there is smoke" is proved. No second opinion, no waiting for more data. (Say it's a perfect alarm that beeps exactly when there's smoke. Real ones also go off at shower steam.)

So your alarm has an odd power. It can prove "there was smoke." It can't prove "there will never be smoke." The same beep that proves the first claim breaks the second, and no run of silence proves it.

That isn't a quirk of smoke alarms. The alarm is a plain case of a result that holds for anything you examine again and again: a detector, a gauge, an experiment you repeat. And the result answers two huge questions at once. What is truth? How can anyone know anything at all?

The short answer

Truth is what holds in reality's record, and knowing a truth is one more event in that record: telling two things apart. Reality keeps a permanent record of every event, because the basic laws of physics never destroy information and the record only grows. Nothing in it is ever erased, and the past still exists. Step 1 gives the reasons in full.

Knowing has an exact shape, and the shape is proved. Take any system you examine again and again. A finite stretch of its readings can do exactly three things:

• Confirm the claims one moment can show: "it happened."
• Refute the claims one moment can break: "it never happens," "it always holds."
• Settle the claims whose answer, yes or no, always shows up at some moment.

Nothing else is in reach. In particular, no measurement ever confirms an exact value, so an exact value can't come from the readings.

Truth also comes with a price list. A truth that needs no added principle sits at the bottom. Every other truth holds given a principle you can name, and "every statement is either true or false" is the most expensive on the list. And a rulebook that checks its claims by rules and contains arithmetic can't prove itself consistent if it is consistent, so trust in it comes from a level above.

Why this matters: two tempting answers to "how can we know anything?" are both wrong. "Nobody can know anything for sure" is wrong, because a finite stretch of readings settles some claims outright and for good, confirms others the moment they happen, and gives every law a verdict at every precision. "Enough data proves anything" is wrong too, because no run of data confirms a "never" or an exact number. The exact list tells you what a test can do for the claim in front of you.

Here's the whole answer in six steps. The rest of this post takes them one at a time and shows how we know each one.

1. Truth is what holds in the record, and a look is an event in it. To know a claim is to look at the record, and the look is one more entry.
2. A finite stretch of readings confirms, refutes and settles exactly three kinds of claim. Two questions block the rest.
3. Settling costs at most a wait, when the settling is proved without "every statement is either true or false." A verdict you can revise reaches further, and it has a limit too.
4. No measurement confirms an exact value. Every law still gets a verdict at every precision.
5. Truth has grades by price. A blank receipt needs no added principle. Every other truth costs a principle with a name.
6. No consistent rulebook of this kind vouches for itself. Trust in it comes from a level above.

Here's the route. I'll stop after each step to zoom out, so you always know where we are and why the next step comes next.

Step 1: Truth is what holds in the record

A claim is true when it holds in reality's record. To know it is to look at the record, and a look is one more event in it.

Here's how we know, starting with what you're doing right now.

The smallest event is one thing told apart from another. Reading this tells one letter from another. Every perception and every measurement tells one thing from another, and the least information that takes is one yes or no. Asking "can I know anything at all?" already uses it: to ask, you have to tell the question from its opposite. So one thing is known before any doubt starts. Things can be told apart.

Two more facts hold in the same act. Comparing two things needs information about both: you can't say "colder" without a note of the earlier sip. And looking at a record is a new event. Footage recorded through noon can't contain the event of someone watching it at one o'clock. It can show a sign that says "this tape will be reviewed," but the sign and the review are different events, and the review comes later.¹ Each of the three holds in the very act of asking whether it holds, so there's no place to stand outside it and doubt it. Turtles All the Way Down? works this out in full.

Reality keeps a permanent record of every event. Every event leaves a complete trace and none is ever erased, so the full record of what happened is part of reality now. The past still exists.

Physics gives a reason: the basic laws never destroy information. In quantum mechanics this is unitarity: the present fixes the past exactly, and running the laws backward recovers every earlier state. It's why physicists treat the black hole information problem as a real problem: losing information would break the laws.²

The record has a reason of its own: it only grows. Each new entry is added with every earlier entry kept, and writing a new entry never alters an earlier one.³ That's separate from what an entry is: each entry is a comparison, and a comparison needs information about both sides.

One step here isn't derived yet: that physical change is this record being written. It's the premise our answers stand on. What's proved is the record's own property, that it keeps every entry.

What a look does. To know that the alarm beeped on day 1,001 is to tell that day from a silent one. That's one distinction, and it goes into the record as one entry. A look at the record is an event like any other, so it adds something the record didn't hold. Can We Ever Know Everything? works that out in its first two steps, and this post starts where that one leaves off.

What's true. A claim about one entry is true when that entry reads that way: "the alarm beeped on day 1,001" is true because the record holds that entry. A claim about every entry, like "the alarm never beeps," reaches past every entry written so far: one entry can break it, and no stretch of entries shows it.

So here's the question that decides how much can be known, and it has an exact answer: what can a finite stretch of readings show about a claim that reaches past it?

Zoom out. Step 1 is done. Truth belongs to the record, and knowing is a look at it. Next, and why: a "never" claim reaches past any look, and any look covers a finite stretch. What can a finite stretch do?

Step 2: A finite stretch of readings confirms, refutes and settles exactly three kinds of claim

Some claims one moment can show. Some claims one moment can break. Some claims both. Those are exactly the claims a finite stretch of readings can confirm, refute and settle, and nothing else is in reach.

Here's how we know, with the alarm and a few words defined.

Check the alarm's log once a day, again and again: beep or silent. The log's readings, day after day, are a run. The first n readings are a stretch of it, a finite part of reality's record. A rule reads a stretch and answers "yes," "no," or "read on." It can be any rule at all, as clever and as fast as you like. This is about information, not machines: no device is described and no deadline is promised.

A rule confirms a claim when it answers "yes" on exactly the runs where the claim is true. It refutes a claim when it answers "no" on exactly the runs where the claim is false. It settles a claim when it does both, so every run gets a right answer, yes or no.

Three claims about the alarm show all three:

• "The alarm beeps on some day." One beep confirms it. No stretch of silence refutes it, because the next day can beep.
• "The alarm never beeps." One beep refutes it. No stretch of silence confirms it.
• "The alarm beeps in the first week." After seven days you know, yes or no. Settled.

Some claims sit outside both circles. Take "the alarm beeps exactly once, ever." It's two halves stuck together. "At least once" can be confirmed and "at most once" can be refuted, but the whole can be neither: a stretch with one beep can still be followed by a second, and a silent stretch can still be followed by a lone beep.

Why exactly these three? A rule answers "yes" after looking at some finite stretch, and every run that begins with that stretch gets the same "yes." So a claim can be confirmed only if, on every run where it's true, some finite stretch already guarantees it. That's what "at some moment, X shows" means. And any claim of that kind is confirmed by the rule that says "yes" at the first stretch that guarantees it. Refuting is the mirror image: one moment where X fails breaks a claim of the form "at every moment, X holds," and only such claims. A claim with both forms is settled.

This holds for every system and every claim. It's Theorem 17.13 of the institute's book The δ Foundation,⁴ and it's the characterization found in formal learning theory (Kevin Kelly, The Logic of Reliable Inquiry, 1996). The book proves it again for its own records and adds the prices in the steps below.

For yes-or-no readings, with every run possible, the settled claims have a simple shape. A claim is settled exactly when some length N, fixed in advance, decides it: any two runs that agree on their first N readings get the same answer. "The alarm beeps in the first week" is one, with N = 7.

What blocks settling. Two questions, and only two. The zero test asks whether a run is silent forever. The search asks whether a run has a beep somewhere. Every claim that no stretch can confirm has the zero test hidden inside it, and every claim that no stretch can refute has the search hidden inside it. Here's how to see it for "the alarm beeps exactly once."

Take any run and build a new alarm from it, so that each day of the new alarm depends on only finitely many days of the run. First way: the new alarm beeps on day 1, and again one day after the run's first beep, if the run has one. It beeps exactly once precisely when the run is silent forever. So a rule that confirmed "exactly once" would confirm "silent forever," and no stretch can do that. Second way: the new alarm beeps only on the day of the run's first beep. It beeps exactly once precisely when the run has a beep somewhere. So a rule that refuted "exactly once" would refute "there's a beep somewhere," and no stretch can do that either.

The exact statement: a claim is settled exactly when neither question is hidden inside it. These are the two names from Is Math Invented or Discovered?, where each is a price a proof has to pay.

Zoom out. Step 2 is done. A finite stretch confirms what one moment can show, refutes what one moment can break, and settles what both can. Next, and why: some settled claims still can't be settled quickly. What does settling cost?

Step 3: What settling costs, and how far a revisable verdict reaches

When a rule is proved to settle a claim without "every statement is either true or false," deciding the claim costs at most waiting. For some claims the waiting has no deadline fixed in advance, but it never takes more information than the record holds. A verdict you can revise reaches claims no stretch ever settles, and it has a limit too.

Here's how we know. In the institute's book the cost of an answer is the principle you have to accept to reach it, and the principle's name is its price. Step 5 lists them all.

Promise that the alarm will beep at least once, but not when. Now take the claim "the first beep falls on an even-numbered day."

A rule settles it: read on until the first beep, then answer by the day's number, even yes, odd no. It's right on every run the promise allows. But it has no deadline. Say some rule promises to answer by day 1,000 at the latest. House A has its first beep on day 1,001, and House B has its first on day 1,002. For the first 1,000 days the two houses are identical, so the rule sees the same stretch and gives the same answer. But the claim is false in A (1,001 is odd) and true in B. The same trick beats any deadline you name.

So the information is all in the record. What can't be known in advance is how long it takes to arrive.

This cost is exact. Call it the wait: the guarantee that a search which can't fail succeeds. For the alarm: if it's impossible that no beep ever comes, then a beep comes, and a search finds it.

Here's the exact result. Suppose someone proves, without "every statement is either true or false," that a rule settles a claim. On any run the rule must answer at some stretch, or the claim would be neither true nor false there. The wait turns that into "the rule does answer at some stretch," a search finds the stretch, and the answer decides the claim. So the wait is enough to decide the claim on every run. And one settled claim costs exactly the wait: the even-day claim above. Not every settled claim costs that much. A claim decided outright, like "the first reading is a beep," costs nothing.⁵

Waiting can pay for the wait, and for none of the other prices. No rule settles the zero test or the search, and answering either one for every run costs its own price. A third question defeats every rule too, the fork. Promise the alarm beeps at most once, ever. Ask a rule to call every such alarm, either "no beep falls on an even day" or "no beep falls on an odd day," and to be right every time. On a silent alarm both calls are right, but the rule must commit by some day n. Put the lone beep just past day n, on the side it ruled out. The stretch is the same, so the call is the same, and now it's wrong.

Now let the experiment change its mind. After every reading it posts a verdict, and it's allowed to flip it later. It decides in the limit when its verdicts end on the right answer and stay there, on every run. It confirms in the limit when its verdicts end on "yes" and stay there exactly on the runs where the claim is true.⁶

Take "the alarm beeps on some day," which no stretch settles. Use this rule: say "no" until a beep, then say "yes" forever. If a beep ever comes, the verdicts end on "yes." If none ever does, they stay on "no." You're always right in the end. What you can't do is tell, on any day, whether the end has come: the "no" on day 1,000 isn't final.

Revising reaches exactly this far. An experiment that's allowed to revise can confirm in the limit exactly the claims of the form "for some m, X_m always holds," like "for some day m, no beep comes after day m." It can decide in the limit exactly the claims where both the claim and its opposite have that form. Every settled claim is decided in the limit, and "the alarm beeps on some day" shows the reverse fails.

And it stops. Take "the beeping dies out," the claim just quoted. A revising experiment confirms it in the limit: say "yes" if today was silent and "no" if it beeped. If the beeping dies out, the verdicts end on "yes" for good. If it never dies out, the verdicts never end on "yes," but they can flip forever instead of ending on "no."

No rule does better. Here's the trick that beats every one. Keep the alarm silent until the rule says "yes, it dies out." Then let it beep once, and go silent again until the rule says "yes" again. Keep going. The alarm now beeps forever, so a rule that decides in the limit has to end on "no." But the rule said "yes" over and over along the way, so its verdicts never end on "no." The same holds for "this orbit stays bounded," when an orbit is read as a stream of recorded positions that can be any stream of whole numbers. That reading is a modelling assumption the proof takes as given.

Zoom out. Step 3 is done. Settling, proved without "true or false," costs at most a wait, and for some claims the wait has no deadline fixed in advance. A verdict you can revise reaches further, and it also stops. Next, and why: all of this is about yes-or-no claims. What about a number, like a weight?

Step 4: No measurement confirms an exact value

A measurement can refute "the value is exactly c." It can never confirm it.

Here's how we know, with a gold bar. The seller swears it weighs exactly 1,000 grams. You weigh it again and again. Each reading is twice as sharp as the one before: the 1st is good to within half a kilogram, the 2nd within a quarter, and the n-th within 1/2ⁿ kilogram. That's the measurement model: a value reaches us as readings with margins that keep shrinking. Two things in this step are named identifications that the proofs take as given, not derived results: that a measurement reaches us this way, and that a physical quantity is a point on the number line.⁷

Suppose the bar secretly weighs 1,001 g. Some of your readings:

Each reading leaves a window of weights that fit it, and a bar fits the readings only if it sits in every window. The 10th reading catches the seller. It says 1,001.4 g and is good to within 0.98 g, so no bar of exactly 1,000 g can give it, and none can give the readings after it.

Now the honest seller, whose bar weighs exactly 1,000 g. Say your scale reads 1,000 g every time. No reading ever refutes that. Does that prove it? No. After the 12th reading, a bar of 1,000.2 g still fits all of them. After the 20th, a bar of 1,000.0005 g does. However many readings you take, a bar a hair heavier still fits all of them, and the hair shrinks without ever reaching zero. The honest seller is never caught. He's never cleared either.

Here's the exact statement. A record of readings fits exactly the values inside all of its windows, and if 1,000 g fits, so does every weight in some small stretch around it. So for any number c, "the value is exactly c" is refutable, never confirmable, and never settled. It's the zero test again, in a number's clothes. Build a number from a run of beeps: it's exactly zero when the run is silent forever, and only then, as Is Math Invented or Discovered? showed.

A derived number gets the same treatment. Take the golden ratio, φ = 1.618..., the positive solution of x × x = x + 1. Any readings that fit φ also fit φ + 1/φ^K for every large enough K (how large depends on how many readings): a different number for each K, and every one different from φ. Agreement between readings and a derived value never separates it from its neighbors. A measurement can refute the claim that a quantity equals a derived value. It can't confirm it. That holds for every derived number, ours included. The result is about numbers and readings. It doesn't say any quantity in nature equals φ.

Where does an exact value come from? Not from a record. It comes from a derivation: φ comes from an equation, with no measurement anywhere in it. The exact numbers in our units come the same way, derived from definitions. The speed of light is exactly 299,792,458 metres per second because the metre is defined from it, and since 2019 the kilogram has been defined by fixing Planck's constant at exactly 6.62607015 × 10^-34 joule-seconds.⁸ Those numbers follow from how the units are defined, not from a measurement that came out exact.

Every law still gets a verdict at every precision. A law is built from equations, "at most" and "less than," joined by "and" and "or." "The bar is exactly 1,000 g" is the simplest. Pick a tolerance, say one gram. A test at that tolerance always has an outcome: the law passes (the bar is within a gram of 1,000), or it's refuted (the readings show a difference, with room to spare). At least one of the two always happens, for any law built this way, and it takes no extra principle.⁹

A law made only of equations and "at most," joined by "and," holds exactly when it passes at every tolerance. That makes it a never-claim: one failed test refutes it, and no run of passed tests confirms it. A strict law, made of "less than," holds exactly when one finite test shows it. So an experiment gives the verdict at a tolerance, refutes an equation or an "at most" law when it's false, and confirms a strict inequality when it's true. It never confirms an equation exactly.

What costs is exactness. Deciding for every pair of numbers whether x = y costs the zero test. Deciding whether x < y costs the search. Turning "not exactly equal" into a visible difference costs the wait. And "of two numbers, one is at most the other" passes every test at every precision, yet asserting it for every pair of numbers is exactly as strong as the fork.

Zoom out. Step 4 is done. A reading can break an exact value and never prove it, so an exact value comes from a derivation, never a record. Every law still gets a verdict at every precision. Next, and why: "costs" has come up four times now: the zero test, the search, the wait, the fork. What is the whole price list?

Step 5: Truth has grades by price

The best truth needs no added principle. Every other truth holds given a principle you can name, and "every statement is either true or false" is the most expensive principle there is.

Here's how we know, starting with one "never" that is known for sure: no count is one more than itself. That claim is about every count, so checking cases one by one can never confirm it. It's derived instead, from the rules of counting in Is Math Invented or Discovered?. The start isn't one more than anything, so it isn't one more than itself. If a count isn't one more than itself, neither is the next, because two counts that step to the same next count were the same count. And every count is reached from the start.

Every proof comes with a receipt, the list of extra principles it leaned on. This proof's receipt is blank, like the receipt for 1 + 1 = 2: it leans on nothing but the rules of counting. A truth like that holds for the counting numbers however the counting is written down, as numbers or as lists of marks.¹⁰ Every other truth holds given a principle with a name.

Each price buys everything joined below it: "every statement is either true or false" gives the search, the search gives the zero test and the wait, and the zero test gives the fork. No two of the six prices are the same: for any two there's a consistent world where one holds and the other fails.¹¹ At the top sits "every statement is either true or false," the principle everybody treats as free, and it's the most expensive of them. Ordinary arithmetic doesn't need it for claims of the form "for every x there's a y with a property you can check." If classical arithmetic proves one, it can be proved without that principle, and the proof comes with a program that finds y.¹² Every number x is y + y or y + y + 1 for some y. The program returns half of x, rounded down: for x = 7 it returns 3, and 7 = 3 + 3 + 1. The full price menu is in Is Math Invented or Discovered?.

Zoom out. Step 5 is done. A blank receipt sits at the bottom, a named principle prices everything else, and "every statement is either true or false" is at the top. Next, and why: one question is left. Can a rulebook vouch for itself?

Step 6: No consistent rulebook of this kind vouches for itself

If a rulebook that checks its claims by rules and contains arithmetic never contradicts itself, it can't prove that it never does. Trust in it comes from a level above.

Here's how we know. Kurt Gödel showed in 1931 that arithmetic, and arithmetic with any finite list of added premises, can't prove its own consistency if it is consistent. The institute's book proves it again for its own proof checker.¹³ M. H. Löb found the same wall for single claims in 1955: a rulebook like this certifies "if I prove φ, then φ is true" only for the claims φ it proves outright.

The way up is a tower, of the kind Alan Turing studied in 1939. Rulebook 0 is arithmetic, and rulebook k + 1 is rulebook k plus the premise "rulebook k never contradicts itself." Every level is consistent. Each level proves the consistency of the level below. No level proves its own, though it's true.¹⁴

Every rulebook can vouch for the one below it. None can vouch for itself. So somebody always has to stand one floor up. Even the proofs of these facts stand one floor up: they're written in ordinary classical mathematics, above the whole tower. Turtles All the Way Down? works out the tower and why it has no top.

Zoom out. Step 6 is done. Every level is consistent, each vouches for the one below, and none for itself. All six steps are in place.

What this changes

You can know some things for sure. Anything one moment shows, once it has happened, because the record keeps it. Anything a fixed opening stretch decides. The verdict of any law at any precision you choose. Every truth with a blank receipt.

What a run of data can't do is confirm a "never," an "always" or an exact number. It can break them. So when someone says "proven," you can ask which kind of knowing they mean: a moment that showed it, a stretch that decided it, a verdict at a stated precision, or a derivation from a principle with a name.

And knowing is never a view from outside the record. It's one more event in it, the same act you started with: telling two things apart, again and again, and writing each one down.

What's still open

• That physical change is the writing of reality's record. The record keeps every entry: that's proved. That evolution is this record being written is the premise our answers stand on, and it isn't derived yet. (Step 1)
• The measurement model and the number line. That a measurement reaches us as readings with shrinking margins, and that a physical quantity is a point on the number line, are named identifications. The proofs take them as given. (Step 4)

The whole trip, in one breath

Truth is what holds in reality's permanent record, and knowing is one more event in it: a look, one thing told apart from another. A finite stretch of looks confirms exactly the claims one moment can show, refutes exactly the claims one moment can break, and settles exactly the claims whose answer, yes or no, always shows at some moment. Settling, proved without "true or false," costs at most a wait, which needs no extra information and for some claims has no deadline fixed in advance, and letting a verdict change reaches further and then stops. No reading confirms an exact value, because a window of readings always holds a rival a hair away, so an exact value never comes from a record, and every law still gets a verdict at every precision. A truth with a blank receipt needs no added principle, every other truth holds given a principle with a name, and "every statement is either true or false" is the most expensive of them. No consistent rulebook of this kind vouches for itself, so trust comes from a level above. That's how knowing is built into reality. One beep proves "smoke" for good. A thousand silent days can't prove "never." A "never" that is known for good is known by derivation, from what distinction forces or from a principle with a name.