Is the Universe Infinite? Only the Way Counting Is.

Here's a library that adds one book every day.

Ask the librarian how many books it has and you get a plain number. Say the library opened 7,300 days ago, about twenty years: 7,300 books. Ask when it stops adding books and you get a different kind of answer: never.

So is the library finite or infinite?

Both answers feel wrong. 7,300 is a number you can write on a napkin. "Never stops" is the biggest idea there is. Hold on to the library. It's the whole post.

Swap the library for the universe and you get the question people ask when they look up at night. Does it go on forever? Is there an edge? What's past the edge?

The short answer

The universe is infinite the way counting is infinite, and in no other way. Counting never runs out of numbers, and it never contains a number called infinity. Every count you reach is a plain, finite number, and the next one is a step away.

The answer is about one thing: reality's record. Every event leaves a complete trace, and none is ever erased. The full record of what happened is part of reality now. The past still exists.

Here's why. The basic laws of physics never destroy information. In quantum mechanics this is called unitarity: the present fixes the past exactly, and running the laws backward recovers every earlier state. That's why physicists treat the black hole information problem as a real problem: losing information would break the laws. So everything that has happened is still held in how things are now.

What kind of record is it? The smallest thing that can happen is one thing being told apart from another, a single yes or no. You're doing it right now, telling these letters apart. Every measurement and every perception tells one thing from another. So each entry in the record is a comparison. A comparison needs information about both sides, so each new entry is written against what came before. The record is written one entry at a time, and the process that writes it runs in ticks: a tick is one step in the writing of the record. The record has its own reason for keeping every entry, too: it only grows. Each new entry is added with every earlier entry kept, and writing a new entry never alters an earlier one.¹

Looking at the record is itself an event. A look is a comparison, telling one thing from another, and a comparison is an event, so a look is written down as one more entry the record didn't hold. (More on the record and on looks in What You Are, which starts from a cup of coffee.) The library is a drawing of the record: each new book stands for one new entry.

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

1. The record never reaches a last page. Reality's record has a first page. Its writing never stops, and every entry it writes is new, so no page is the last. At any moment the record holds a plain, finite number of entries, and the number keeps growing.
2. Counting has no last number. The record is a count: a blank start, then one more, then one more. Every count has a next one, and every count is a finite number of steps from the start. There is no step called infinity.
3. Nothing real is uncountable. Here "real" means "exists in the world", not "the real numbers" of mathematics. Everything real leaves a finite stretch of the record, and finite stretches can be listed. So every real collection can be listed, one member after another. Nothing real is a finished infinity bigger than counting.
4. No finite record can tell "forever" from "very big." Every look anyone makes takes in a finite stretch of the record. Any finite pile of looks fits under some ceiling, so no pile of looks can show whether there is a ceiling at all.
5. How far space reaches isn't derived yet. And no survey of space can settle it, however long it runs.

Why this matters: "infinite" is the word people grab when they can't picture a size, and it hides two different ideas. One is "never stops": there's always one more. The other is "already finished and bigger than any list": a whole endless line sitting there with every point on it. Reality has the first and not the second. That gives time the shape of counting, a first page and then forever. It means nothing ever happens "after infinite time," because no tick is infinitely far away. And it draws a line around what looking can tell you: no measurement, however good, shows whether something goes on without end or only for a very long time.

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

Step 1: The record never reaches a last page

Reality's record has a first page and no last page. The first page comes from counting. The missing last page rests on one premise, the clock's rule. Call the process that writes the record the clock; each of its ticks is one step in the writing of the record. Its rule: the writing never stops. However many entries have been written, the clock writes another one later.

Here's how we know the record has a first page and no last page. Start with the first page. A record is a count of events that happened, and the least that can have happened is nothing, so the count can't go below zero. That gives the record a floor: a blank first page with nothing written on it yet. (More in Did Time Have a Beginning?.)

Now the other end. Every entry the clock writes is new: a later event is never already in the earlier record. Take the record as it stood just before its seventh entry was written. It can't hold the seventh entry, because that entry is the writing, and the writing hasn't happened yet. Once it's written, the record holds it. The same goes for the eighth entry, the ninth and every one after. So every entry the clock writes makes the record bigger.

A record that grows with every entry, and whose writing never stops, has no last page. There's no tick after which it stops growing.

Page 0 is the blank first page, and pages 1 through 4 are written, one entry to a page. (Did Time Have a Beginning? calls the blank first page "page 1". Here each page carries the number of entries written so far, so the blank page is page 0, just as the count in Step 2 starts at 0.) "Now" is the front edge, the place where the writing is happening (Time is about that edge). Everything past it is blank, and there's always another blank page, so no page can take a THE END stamp. (No last page doesn't say what gets written on the later pages. That's a different question: how the universe ends.)

There's a second way to see that the record has no last page. Every look at the record is a new entry. So a record that keeps looking at itself never finishes listing itself: each look it lists is one more entry to list. It outgrows every finite size. (More in Turtles All the Way Down?.)

Nothing else in the structure forces the clock's rule. A clock that writes through tick 4 and then goes quiet is a consistent world, and its record goes flat from tick 4 on.

Reality has no last page because its clock never stops writing. A last entry would be the one observation that breaks the premise.

Zoom out. Step 1 is done. The record has a first page and no last page, because the clock never stops writing and every entry it writes is new. Next, and why: "no last page" says the count never stops. What kind of count is it? Step 2 looks at the count itself.

Step 2: Counting has no last number

The record is a count: a blank start, then one more, then one more. Every count has a next one, the next one is always new, and every count is a finite number of steps from the start. So there's no last number, and no number called infinity.

Here's how we know. The counting numbers are built from one act: telling two things apart, again and again. The record of that act is the count: a blank start, and one fresh entry each time. (More in Is Math Invented or Discovered?.) Three rules hold for the count, and each one is proved:²

• The start is never "one more" of anything.
• Two different counts never step to the same next count.
• Every count is reached from the start.

Each rule has a plain reason. The start is the blank, and one more of anything holds at least one entry, so the two are never the same. If two counts stepped to the same next count, taking away that newest entry would give back both of them, so they were the same count. And every count was built from the blank, one entry at a time.

Back to the library. Day 1: one book. Day 2: two. Each day's total is reached from the empty shelf by adding one book at a time, so each day's total is a finite number. Day 7,300 has 7,300 books. No day has "infinitely many." And there's no day when the librarian runs out of numbers, because the next day's total is always one more, and it's never a total the library has had before.

Here's the question that comes next. If the count never stops, isn't it infinite?

It never stops, yes. That's the whole of the infinity: there's no last step. It's not a number you can reach, and it's not a finished pile. The third rule carries this. Every count is reached from the start, so every count is a finite climb up the staircase. No step takes forever to reach, and there's no step called infinity.

So here's what "infinite" means for the record. It doesn't name a size. It means there's no last step. No tick is infinitely far from the first page: every tick that ever comes is a finite number of ticks after the first one. So "what happens after infinite time?" has no answer, because there's no such time. Infinity isn't a place the record reaches. It's the fact that the record always has a next page.

And there's only one count. Anything with a start and a step that obeys the three rules is this count in other clothes, lined up with it in exactly one way. A clock that comes back around to 12 breaks the first rule, so it isn't one.

Here's why they line up in exactly one way. Set any such thing beside the count, and pair start with start, then each next with the next. That fixes every pair, so there's only one way to line them up. The third rule means every member gets a partner, and the first two mean no two members share a partner.³

Time has the shape of counting. Finite behind you, forever ahead: a first page, and then one more, and one more.

Zoom out. Step 2 is done. The record is a count with a blank start, no last number and no number called infinity, so every tick is a finite number of steps from the first page. Next, and why: counting is the smallest kind of infinity there is. Textbooks have bigger ones, like the number line, which has more points than any list can hold. Does reality have those?

Step 3: Nothing real is uncountable

Every real collection can be listed, one member after another, with every member getting its own shelf number. Nothing real is a finished infinity bigger than counting.

A note on the word "real". "The real numbers" is a name from mathematics, for the points on the number line. Here "real" means something else: exists in the world. The number line comes back at the end of this step.

Here's how we know. Start with three words.

A list is a shelf with numbered spots: spot 1, spot 2, spot 3, and on. A collection is countable when every member can have its own spot. The counting numbers are countable: number n goes in spot n. The even numbers are countable too, though they're only half the counts: 2 goes in spot 1, 4 in spot 2, 6 in spot 3, and 2n in spot n. Fractions are countable as well, which surprises people. (The math post has the rule that gives each fraction a spot.)

Some collections can't be listed. Take every endless pattern of beats and rests: on each beat, forever, a drum plays or rests. Write a beat as 1 and a rest as 0, and each pattern is an endless row of 0s and 1s. Whatever list of such patterns you write, there's a pattern the list missed. Take the diagonal: at beat 1 do the opposite of what pattern 1 does, at beat 2 the opposite of pattern 2, and so on down the list. The new pattern differs from every pattern on the list. Georg Cantor found this in 1891.⁴ So there are more endless patterns than spots, and the points on the smooth number line are in the same boat. A collection like that is uncountable.

Now, what's real? This step rests on one premise, the way Step 1 rests on the clock's rule: what's real is what's in the record. Here's the reason. Every way anything shows up, whether a measurement, a perception or a collision, is an event that tells one thing from another, and every event is an entry in the record. So a thing is real when it turns up in a finite stretch of the record, and a later look can tell it from every other real thing. The stretch is finite because every tick is a finite number of ticks from the first page (Step 2). That's all it takes to exist in a record: to be written, and to be found. The listing below is a proof from that premise.

Now count the real things. A look can only read the record. So two things that left the same record would get the same answer from every look, and nobody can tell them apart. Different real things therefore leave different records. Each record is a finite string of marks. And finite strings can be listed, the way a dictionary lists words: every one-mark string, then every two-mark string, then every three-mark string, and so on. With two kinds of mark, 0 and 1, that's 2 strings of length one (spots 1 and 2), then 4 of length two (spots 3 to 6), then 8 of length three (spots 7 to 14). The string 0110 has length four, and it's the seventh of the sixteen length-four strings, so it lands in spot 14 + 7 = 21.

Every real thing gets the spot of its own string, and no two things share a spot. That's a list. So every real collection can be listed.

An endless pattern isn't a finite string. It never ends, so it never turns up in the dictionary, and no list at all holds all of them. A collection like that isn't real: no record holds it.

That settles the number line. Its uncountably many points aren't a real collection. Here's what the line costs: every exact test on it takes a power the record doesn't have for free. Take the plainest exact test, telling for any two points whether they're exactly equal. A rule that does that also tells, for any endless row of 0s and 1s, whether a 1 ever turns up. Here's how: write the row after a decimal point, so 0, 0, 1 ... becomes the point 0.001.... That point equals zero exactly when no 1 ever turns up. No finite stretch of record can tell that. Every measurement anyone has made comes out as a finite string of digits. (More in Is Math Invented or Discovered?.) Everything real is in the record, so the full number line is a tool we use and not a thing in the world. It's a very good tool. It's also a finished infinity bigger than counting, and those aren't in the record.

Zoom out. Step 3 is done. Everything real leaves a finite stretch of the record, finite stretches can be listed, and so every real collection can be listed. Nothing real is uncountable. Next, and why: Step 3 says nothing real is bigger than counting. It doesn't say whether counting itself is all there, or whether reality stops at some huge number. Can anyone check by looking?

Step 4: No finite record can tell "forever" from "very big"

Nobody can find out by looking whether there's a limit on how much of the record one look can take in. That's not a shortage of instruments. Any finite record of looks fits both answers, and that's exact.

Here's how we know. Go back to the library and add an inspector with a clipboard.

The inspector wants to know whether the building has room for infinitely many books or only for a huge number. The only check is looking at books, and the inspector has only ever looked at finitely many. Say it's ten million. A building with room for ten million and one books looks exactly the same from where the inspector stands. So does a building with room for a trillion. So does one with room for infinitely many. The inspector's pile of looks fits all of them. If someone names a limit, say exactly ten million, one look at book 10,000,001 settles it. But nobody names the limit, and any finite pile of looks fits under some limit.

Now the exact version. A look is one comparison: telling one thing from another, which needs information about both. A look's reach is how much of the record it takes in. We count reach in bits, where a bit is one mark of the record, a 0 or a 1. There are two ways to read the limit:

• No ceiling fixed in advance. Whatever look you name, some size of record holds it.
• One ceiling for every look. There's a single size B, and no look takes in more than B bits.

The two agree on every finite collection. They differ only on infinite ones. Under a ceiling of B bits, only finitely many records fit in B bits, so only finitely many things can ever be told apart. That number grows with the ceiling and never becomes infinite.⁵ With no ceiling fixed in advance, counting itself is real: the counting numbers are a real collection, infinitely many things told apart.

And here's the proved fact. A finite record of looks is possible in a world with no ceiling exactly when it's possible in a world with some ceiling. The biggest look in the record has a finite size, and a ceiling above that size holds the whole record. So no finite record separates the two readings.

Now the world. There are two exact rules about what nature allows, and they exclude each other, so at most one of them holds:

• No ceiling. After any stretch of record, one more entry can be written and any one comparison can be made, however far back its entries sit. No entry ever goes out of reach.
• A window. Every comparison reaches only the newest N bits of the record, and those N bits are the window. Older entries are out of reach. The record can keep growing and looks can keep happening forever, yet only finitely many things are ever told apart.

They can't both hold. With a window, once enough entries are written the oldest entry is out of reach for good, and "any one comparison" fails. A window of a stated size is easy to catch out, because one recorded look that reaches past it ends it. A ceiling of unstated size isn't. No finite record catches it, for the reason above: any finite record of looks fits under some ceiling.

Whether an entry ever goes out of reach for good is open. No finite record can settle it, and the record's own rules allow both answers. So the answer has to come from physical law: from what physics lets a look reach.

Zoom out. Step 4 is done. Any finite record of looks fits under some ceiling, so no record can tell "no ceiling" from "a ceiling somewhere." The two exact rules, no ceiling and a window, exclude each other. Next, and why: the question you started with. How far does space reach?

Step 5: How far space reaches

How far space reaches isn't derived yet. And no survey of space can settle it, however long it runs.

Here's where things stand.

What's built. Our models of space use a grid of cells in three dimensions. The grid in Did Time Have a Beginning? has a cell at every whole-number position in three directions, so it goes on without end. That's a choice made when the model is built. In other versions of our models the grid is finite and wraps around on itself: walk off the right edge and you come back in on the left, like the edges of an old arcade game's screen. Both kinds appear, and both are put in by hand. Nothing derived picks between them. The questions underneath aren't derived either: where a cell sits, how far apart two cells are, and how far anything can get in one tick. You can't derive how far space reaches before you've derived what "far" means.

What telescopes see. We see only as far as light has had time to travel. Inside that region, space measures flat to within a fraction of a percent. Planck's map of the early universe's light, combined with galaxy surveys, puts the curvature parameter, the number that measures how much space bends overall, at 0.001 plus or minus 0.002, and zero is perfectly flat.⁶ A flat space can go on forever. A space with a bend too gentle to detect can also close up on itself, far past what we can see. If space closed up within what we can see, light from the same patch of the early universe would reach us from two directions, and the map would show pairs of circles with matching patterns. Searches for those matched circles have found none.⁷ The measurements fit both.

Why no survey settles it. Here's the question in its barest form. Survey the universe stage by stage, and at each stage record one thing: did this stage find a region nobody had recorded before? I'll draw the reports as a pattern of beats and rests, like the patterns in Step 3: a beat for "found one" and a rest for "found nothing new." A finite universe gives a pattern with some number of beats and then rests forever, because every region is on the books. An endless one gives a pattern that never stops, with rests of any length between the beats.

Say the survey has found nothing new for a billion stages in a row. Is the universe finite? You can't tell, because the next stage can still find a new region. It's the fact from Step 3 again: no number of rests tells you a beat never comes. And say the survey keeps finding new regions. That doesn't show it never stops either, because a finite universe can have a very long list of regions before it runs out. So no finite stretch of survey confirms "finite," and none confirms "endless."

It gets stronger. Let the survey take its verdict back as often as it likes, as long as it settles in the end. Still no rule works. One rule gets finite universes right in the long run: guess "finite" unless the latest stage found something new. But on an endless universe with rests between the beats, that rule flips every time a new region turns up, and never settles. And no rule of any kind settles on the right answer in every possible universe.

Here's why. A rule that gets some finite universe wrong has failed already, so take any rule that gets every finite universe right in the long run. Feed it stages that find nothing new until it says "finite". It has to say it sooner or later, because if nothing new ever turned up again, the universe would be finite. Then give it one stage that finds a new region, and wait again until it says "finite", which it has to do for the same reason. Do that forever. The universe you've built never stops turning up new regions, so it's endless, and the rule says "finite" again and again. It never settles on "endless". That's proved for a survey modeled this way, as a stream of "found a new region" and "found nothing new" reports. That a real survey is such a stream is the premise under this step: no survey ever sees space close up on itself. The matched-circle searches back it as far as anyone has looked.

So the answer to "does space go on forever?" isn't waiting in a telescope. A survey is a finite record, and Step 4 says what a finite record can't do. The answer has to come from the structure of reality: from what its rules allow space to be.

Zoom out. Step 5 is done. The grids the models run on are put in by hand, telescopes fit both answers, and no survey of any length settles it. Next, and why: all five steps are done, so here's the whole route in one breath.

The whole trip, in one breath

Reality's record has a first page and no last page: its clock never stops writing, and every entry it writes is new. The record is a count, with no last number and no number called infinity, so every tick is a finite number of steps from the first page. Everything real leaves a finite stretch of the record, finite stretches can be listed, and so nothing real is uncountable. That's why the universe is infinite the way counting is infinite, and in no other way. No finite record can tell an endless reach from a huge but capped one, because any finite pile of looks fits under some ceiling. No survey will settle how far space reaches, because a survey is a finite record.

The biggest idea there is turns out to be made of the smallest move there is: one more. That's how reality is built: a first page, then forever, written one entry at a time, in a book with no last page.