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.

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. Every tick of its clock writes one more entry, and every tick has a next one, 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. 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.

Two things from earlier posts do the work here. Reality keeps a permanent record of every event: entries go in one after another and nothing is ever erased. I draw the record as the Ledger, a thick book with a deadpan face and one response to everything: Noted. And looking at the record is itself a new event. The look is one more entry the record didn't hold. Both come from What You Are, which starts from a cup of coffee. The library is a drawing of the record: one book a day stands for one entry per tick.

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. The Ledger keeps the map.

Step 1: The record never reaches a last page

Reality's record has a first page and no last page. Every tick of the clock writes one entry, and there's always a next tick, so there's always a next entry.

Here's how we know. Start with the first page. Did Time Have a Beginning? shows it: a record is a count of events that happened, and a count of events that happened can't go below zero. So the record has a floor, a blank first page with nothing written on it yet.

Now the other end. The process that writes the record runs in ticks, and every tick writes one entry: what each clock did and which way the flow went. That's the clock's rule. One entry at every tick, and a tick after every tick.

Each of those entries is new. It's the third fact from What You Are again: a later event is never already in the earlier record. Take the record as it stood before tick 7. It can't hold the entry tick 7 writes, because that entry is the writing, and the writing hasn't happened yet. After tick 7 the record holds it. The same goes for tick 8, tick 9 and every tick after. The record gains an entry at every tick.

A record that gains an entry at every tick has no last page. There's no tick when it stops gaining.

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. The Ledger has a pen, a lot of paper and no THE END stamp. It checked. (No last page doesn't say what gets written on the later pages. That's a different question: how the universe ends.)

Everything in this step rests on one premise: the clock writes an entry at every tick. Nothing else in the structure forces it to. 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 tick would be the one observation that breaks the premise.

There's a second way to see it, from Turtles All the Way Down? The record includes its own looks, and every look at the record is a new entry. A record that keeps looking at itself can never finish listing itself, so it outgrows every finite size.

Zoom out. Step 1 is done. The record has a first page and no last page, because every tick writes one new entry and the clock never stops. 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. Is Math Invented or Discovered? builds the counting numbers 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. Three rules hold for it, and each one is proved:¹

• 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.

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.

Yeah-But, the voice in the back of your head that has never once said wow, clears his throat. "Yeah, but if it never stops, it's infinite. You said so."

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.

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.

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. Gary, the coin-flipping drummer from earlier posts, plays or rests on every beat, forever. Whatever list of endless drum patterns he writes, 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 1891.² 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? In the record, 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.³ That's all it takes to exist in a record: to be written, and to be found.

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. Is Math Invented or Discovered? shows what the line costs: every exact test on it takes a power the record doesn't have for free. 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. Yeah-But will do, with a clipboard.

He wants to know whether the building has room for infinitely many books or only for a huge number. He can only check by looking at books, and he 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 he stands. So does a building with room for a trillion. So does one with room for infinitely many. His 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. (From Part 1: comparing two things needs information about both.) A look's reach is how much of the record it takes in, counted in bits, where a bit is one mark, 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 things can ever be told apart, because only so many different records fit in B bits: at most (B + 1) times 2 to the power B, plus 1. That's a generous limit, not an exact count, and it's finite. For a ceiling of 3 bits that's at most 4 × 8 + 1 = 33 things. For 10 bits it's at most 11 × 1,024 + 1 = 11,265. The number grows with the ceiling and never becomes infinite. 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:

• 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.
• A window. This is a ceiling of one particular shape: every comparison reaches only the newest N bits of the record. 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.

Which rule nature follows is open. No finite record can settle it, so the answer has to come from the structure of reality: from what its rules allow a look to 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. The first world with clocks and light is the grid from Did Time Have a Beginning?: a three-dimensional grid of cells, four clocks in each. The grid that post uses 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 at 0.001 plus or minus 0.002, and zero is perfectly flat.⁴ 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. 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 Gary's drumming: a beat for "found one" and a rest for "found nothing new." A finite universe is a Gary who plays some number of beats and then goes silent forever, because every region is on the books. An endless one is a Gary who never stops playing, 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 the math post: no number of rests tells you Gary never plays again. 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. That's proved for a survey modeled this way, as a stream of "found a new region" and "found nothing new" reports.

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 writes one entry at every tick. 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.

(The Ledger would like you to know that it holds a perfectly ordinary number of entries right now. It would tell you which one, but the number would be out of date before you finished reading it. Noted.)