What Is the Biggest Number — And Why Infinity Breaks Mathematics

Ask a group of children to name the biggest number and the game escalates fast: a million, a billion, a trillion, "a trillion trillion," and then someone shouts "infinity plus one" and the argument starts. The playground version is closer to real mathematics than it looks. There is no biggest number, because you can always add one. But the contest to name huge numbers is real, it has serious champions, and past all of them sits infinity, which turns out to come in different sizes. What follows walks the whole ladder, from a googol to Graham's number, then to Cantor's proof that some infinities outrun others.

Key takeaways

• There is no largest finite number: for any whole number n, the number n + 1 is larger
• A googol is 10^100, and a googolplex has more digits than there are particles in the observable universe, so it cannot be written out in full
• Graham's number is built with Knuth's up-arrow notation iterated 64 times, and TREE(3) exceeds it
• Infinity is not an ordinary number, but Georg Cantor showed in 1874 that the real numbers form a strictly larger infinity than the counting numbers
• Whether any infinity sits between those two sizes (the continuum hypothesis) cannot be settled from the standard axioms of set theory

Why there is no biggest number

Take any whole number you like and add one. The result is bigger, and it is also a whole number, so you are never done. This is not a clever trick. It follows from the way the counting numbers are defined: every number has a successor, and the successor is never the number you started with.

People have tried to outrun this for a long time. In the third century BCE, Archimedes wrote The Sand Reckoner to answer a Greek claim that the grains of sand were beyond counting. He built a number system based on "myriad myriads" (10^8), estimated the size of the universe using the heliocentric model of Aristarchus, and concluded that no more than about 10^63 grains of sand would fill it. According to Wikipedia's summary of the work, the largest number he named was 10^(8 × 10^16). The point was not the sand. It was proof that a good notation can name numbers that seem to defeat counting.

That is the key to every contest since. As the computer scientist Scott Aaronson puts it in his essay "Who Can Name the Bigger Number?", "Place value, exponentials, stacked exponentials: each can express boundlessly big numbers," but they differ in how concisely they do it. The winner of a biggest-number contest is whoever owns the deeper notation, not the fastest pen.

How big is "big"? A ladder of finite numbers

Humans handle numbers up to a few thousand intuitively and then lose the thread. A quick way to feel the difference between a million, a billion, and a trillion is to count seconds.

Quantity	Written as	Feels like
One million seconds	10^6	about 11.6 days
One billion seconds	10^9	about 31.7 years
One trillion seconds	10^12	about 31,700 years, far longer than recorded history
Hydrogen atoms in the observable universe	about 10^80	The "Eddington number"
A googol	10^100	Bigger than the atom count by a factor of about 10^20
A googolplex	10^(10^100)	More digits than there are atoms

The three seconds figures follow from simple division, using 86,400 seconds in a day and about 31.6 million in a year. The atom figure comes from the Wikipedia article on the observable universe, which gives approximately 10^80 hydrogen atoms (the Eddington number), derived by dividing the mass of ordinary matter by the mass of one hydrogen atom. If you are curious how large that universe is, see how big the universe is.

Growth by repeated multiplication is the reason the later rows explode. Money works the same way at small scale, and compound interest is the everyday glimpse of exponential growth. A googol is just the same trick applied until the exponent is 100.

Googol and googolplex

The word "googol" came from a child. According to Wikipedia's entry, Milton Sirotta, a nine-year-old and nephew of the mathematician Edward Kasner, coined it in 1920, and Kasner popularized it in Mathematics and the Imagination (1940). A googol is a 1 followed by 100 zeros, so it has 101 digits. The search company Google took its name from a deliberate misspelling.

How large is that? Carl Sagan noted that the observable universe holds about 10^80 elementary particles. Even if the universe were packed solid with neutrons, the same page notes, the count would be about 10^128, which is still far short of a googol. A googol is not a number that counts anything physical.

A googolplex is defined as 10 raised to the power of a googol, or 10^(10^100). Its decimal form is a 1 followed by a googol of zeros. Here the notation matters: the number of zeros alone is 10^100, which already exceeds the roughly 10^80 atoms available to write them on. You could not write out a googolplex even if every atom in the universe were a pen and a digit. And yet it is tiny by the standards of what comes next.

Why exponent towers need a new notation

Exponents let us write a googolplex in a few characters. To go further, we need an operation that is to exponents what exponents are to multiplication. In 1976, Donald Knuth proposed one, and it is now called up-arrow notation. The Wikipedia article on Knuth's up-arrow notation lays out the rules:

• One arrow is ordinary exponentiation: 2 ↑ 4 = 2^4 = 16
• Two arrows is tetration, or repeated exponentiation: 3 ↑↑ 3 = 3^(3^3) = 3^27 = 7,625,597,484,987
• Three arrows is pentation, or repeated tetration: 3 ↑↑↑ 3 is a tower of 3s whose height is 7,625,597,484,987

Pause on that last line. A tower of exponents a few dozen levels tall already dwarfs a googolplex. A tower more than seven trillion levels tall is beyond any physical description. And it is only the result of three 3s joined by three arrows.

Think of it as a staircase where each step is a different kind of machine. Addition is the first step, multiplication repeats addition, exponents repeat multiplication, and each new arrow repeats the one before. The jump between steps is not a bigger number. It is a faster way to grow.

Graham's number

Graham's number arose from a problem in Ramsey theory, a branch of combinatorics that studies when order is forced to appear in large systems. It concerns coloring the lines that join corners of a high-dimensional cube, and the number was an upper bound on how many dimensions are needed before a certain pattern is guaranteed. According to Wikipedia, Ronald Graham described it to Martin Gardner, who popularized it in Scientific American in 1977, and the 1980 Guinness Book of World Records listed it.

The definition is a staircase built on top of a staircase. Let g₁ = 3 ↑↑↑↑ 3, using four arrows. Then g₂ has g₁ arrows between its 3s, g₃ has g₂ arrows, and so on. Graham's number is g₆₄. Each layer takes the whole previous number and uses it as the count of arrows. To picture it, imagine a skyscraper where each floor is designed by the entire floor below, and there are 64 floors.

The scale is hard to state without sounding glib. The same Wikipedia page says that the observable universe is "far too small to contain its ordinary digital representation," even if each digit occupied a single Planck volume, and that the number of digits is itself too large to represent there. Yet we know things about it. Its last ten digits are 2464195387, so parts of it can be computed even though the whole cannot.

Here is a detail worth knowing. Graham's number was only an upper bound, a ceiling to guarantee the proof worked. The true answer is far smaller. The same article gives current bounds of 13 ≤ N* ≤ N'', with the lower bound from Jerome Barkley (2008) and the improved upper bound from Eryk Lipka (2019), so the quantity that Graham's number once capped is now known to lie between 13 and a much smaller, though still enormous, upper limit. Mathematicians often use a monstrous number as scaffolding to be taken down later.

Numbers that show up on their own: Skewes' number

Not every huge number is a stunt. Some appeared because a proof needed a safety margin. The prime-counting function π(x) counts primes up to x, and the logarithmic integral li(x) is a smooth formula that approximates it. For every value anyone has computed, li(x) is larger. Littlewood proved in 1914 that the order eventually flips, but he gave no bound on where.

In 1933, Stanley Skewes showed, assuming the Riemann hypothesis, that a flip happens below e^(e^(e^79)). According to Wikipedia's article on Skewes's number, a 1955 bound that does not need that assumption was e^(e^(e^(e^7.705))). The 1933 figure is vastly larger than a googolplex. Later computation using zeros of the zeta function shrank the bound, and Saouter, Demichel, and Zegowitz (2010) narrowed it to roughly e^727.95, about 1.397 × 10^316. No crossing has been found below 10^19.

The pattern repeats: a theorem needs a bound, the first bound is absurdly large, and better methods pull it down. The numbers shrink as understanding grows, and the biggest ones are often monuments to what we did not yet know.

TREE(3) and the Busy Beaver

TREE(3) does not need a stunt either. The function comes from a result called Kruskal's tree theorem, which the Wikipedia article says was conjectured by Andrew Vázsonyi and proved by Joseph Kruskal in 1960. Harvey Friedman defined the TREE function in the early 1980s. Roughly, you try to build a sequence of labeled trees, where the first has at most one vertex, the second at most two, and so on, such that no tree is contained inside a later one. TREE(n) is the longest such sequence you can build using n labels.

The first values are small: TREE(1) = 1 and TREE(2) = 3. TREE(3) is a different kind of object. The page calls it "one of the largest simply defined finite numbers," and says that before TREE(3) grows to its size, numbers like Graham's are "surpassed by comparison." One very weak lower bound is A^(A(187196))(1), where A is the Ackermann function, a function so fast-growing that it already outruns every towers-of-exponents scheme. The sequence of values is finite for every n, since a finite version of the theorem can be proven, but the proof that it must end lives in strong mathematical systems.

If you want a larger contender with a different flavor, there is the Busy Beaver function, introduced by Tibor Radó in 1962. It asks: among all Turing machines (abstract computers) with n internal states that eventually stop, how long can the longest-running one run? The values are known only for the smallest n.

n	Longest run S(n), in steps	Most 1s written Σ(n)
1	1	1
2	6	4
3	21	6
4	107	13
5	47,176,870	4,098
6	greater than 2 pentated to the 5	greater than 2 ↑↑↑ 5

The 2024 proof that S(5) = 47,176,870 was carried out by an online collective of amateurs and formally verified in the Rocq proof assistant, according to Wikipedia's page on the busy beaver. The jump to six states already lands in up-arrow territory.

What makes the Busy Beaver unusual is why it grows so fast. Aaronson's essay explains that if any computable function grew faster than it, you could use that function to solve the halting problem (the question of whether an arbitrary program ever stops), which Alan Turing proved impossible. In Aaronson's words, "The Busy Beaver sequence is non-computable, solely because it grows stupendously fast." Every function you can write a program for is eventually left behind. Wikipedia adds that a 25-state machine exists that halts only if it finds a counterexample to Goldbach's conjecture, so knowing S(25) would settle an open problem, though that number is far too large to write down.

Infinity is not a bigger number

Everything so far has been finite. Infinity is something else. In ordinary arithmetic, you cannot treat it as a number: adding one to it changes nothing, and subtracting it from itself has no definite answer. It is better described as a property, the property of having no end, which different branches of math define in different ways.

Cantor's key idea was to stop asking how big a collection is and ask how to compare collections. Two sets have the same size if you can pair every element of one with exactly one element of the other, with nothing left over. This pairing is called a one-to-one correspondence, or bijection. For finite sets, it is the same as counting. For infinite sets, it overturns your intuition.

Think of a theater with a seat for every guest and no guest left standing. If the seats and the guests can be paired up perfectly, the two groups are the same size, even if you never learn how many there are. Cantor made that test the definition of size.

Hilbert's hotel and the same-size trap

The best-known illustration is a thought experiment. According to Wikipedia's article on Hilbert's paradox of the Grand Hotel, David Hilbert introduced it in a 1924 to 1925 lecture and George Gamow popularized it in his 1947 book One Two Three... Infinity. The hotel has infinitely many rooms, numbered 1, 2, 3, and every room is occupied.

A new guest arrives. The manager asks every guest to move from room n to room n + 1, which frees room 1. Infinitely many new guests arrive? Move each guest from room n to room 2n, which frees every odd-numbered room, and there are infinitely many of those. A full hotel can still welcome more people, which is exactly what never works with finite hotels.

The same logic shows that the counting numbers (1, 2, 3, ...) and the even numbers (2, 4, 6, ...) are the same size, even though the evens are only half the counting numbers. Pair 1 with 2, 2 with 4, 3 with 6, and so on. Nothing is left over on either side. Cantor named this size ℵ₀ (aleph-null), and a set of that size is called countably infinite. The whole numbers, the integers, and the fractions are all countable.

Cantor's diagonal argument

After this, you might expect every infinity to be the same size. In 1874, Cantor proved otherwise. His first proof that the real numbers (all the points on a number line, including irrationals such as π and √2) cannot be matched with the counting numbers appeared that year, and the MacTutor biography records it. A simpler proof, the diagonal argument, appeared in 1891, per Wikipedia.

Here is the argument in plain steps. Suppose someone claims to have an infinite list of every real number between 0 and 1, written as decimals: r₁, r₂, r₃, and so on. Build a new number like this:

1. Look at the first digit of r₁ and change it
2. Look at the second digit of r₂ and change it
3. Look at the third digit of r₃ and change it
4. Continue down the diagonal forever

The new number differs from r₁ in the first place, from r₂ in the second, from every listed number in at least one place. So it is not on the list, and the list was not complete. (One technicality: some numbers have two decimal forms, such as 0.5 and 0.4999..., so you choose replacement digits that avoid 0 and 9 to dodge the problem.)

Picture a rigged raffle. Someone announces that every ticket is in the drum. You walk down the drawing sheet and write a number that disagrees with the first ticket in the first digit, the second ticket in the second digit, and so on. Your number cannot match any ticket in the drum, so the claim that all tickets are present is false. The same trick defeats every list.

Therefore the real numbers cannot be listed. They form an uncountable infinity, strictly larger than ℵ₀. Their size is called the cardinality of the continuum, written 𝔠 or 2^ℵ₀.

Infinitely many infinities

The trick generalizes. Cantor's theorem says that for any set S, the set of all its subsets (the power set) is strictly larger than S. The proof uses the same self-reference as the diagonal. Given any pairing f from S to its subsets, define T as the set of all elements s that are not in the subset f(s). If s is in T, then s is not in f(s), so T differs from f(s). If s is not in T, then s is in f(s), and again T differs from f(s). Either way, T is a subset that the pairing missed.

Apply it again and again. The power set of the counting numbers has the size of the real numbers. The power set of the reals is bigger still, and the power set of that is bigger again. There is no top rung: infinity comes in an unending tower of sizes. Cantor labeled the first few cardinal numbers ℵ₀, ℵ₁, ℵ₂, and so on, where ℵ₁ is the next size up from ℵ₀, as the Wikipedia article on aleph numbers explains.

So the playground argument gets a verdict. "Infinity plus one" is not bigger than infinity as a cardinal: ℵ₀ + 1 = ℵ₀, as the hotel suggests. In Cantor's other system, the ordinals, which record order instead of size, ω + 1 is a genuinely different number from ω, but the set it describes is still countable and has size ℵ₀. The child who said "infinity plus one" was partly right and partly wrong, depending on which kind of number you meant.

The continuum hypothesis: a question with no answer

An obvious follow-up question is whether anything fits between the counting numbers and the real numbers. The continuum hypothesis (CH) says no: that the size of the reals is the very next infinity after ℵ₀, so 2^ℵ₀ = ℵ₁. Hilbert placed it first on his 1900 list of problems for the twentieth century. The MacTutor biography notes that Cantor himself believed he had proved it more than once, and found an error each time.

The resolution was strange. According to the Stanford Encyclopedia of Philosophy, Kurt Gödel showed in 1938 that CH holds in a particular mathematical universe, the constructible universe, so it cannot be disproved from the standard axioms (ZFC). In 1963, Paul Cohen introduced a technique called forcing and showed that CH also cannot be proved. The hypothesis is independent of ZFC: the axioms permit it to be true and permit it to be false. See also the profile of Kurt Gödel, who proved much deeper limits on formal systems.

Compare it to geometry. For centuries mathematicians tried to prove Euclid's parallel postulate from his other axioms, and the result was the discovery that you can build consistent geometries where it is true and where it is false. CH is similar. It does not mean the question is meaningless. Some set theorists, working with new axioms based on large cardinals, argue for a definite answer. The same Stanford entry notes that under one such axiom of Hugh Woodin, the continuum has size ℵ₂, though his case is disputed and a parallel case for CH exists. The matter remains open as a question of which axioms to adopt.

A paradise that people tried to evict

Cantor's results provoked fierce opposition. According to Wikipedia, Leopold Kronecker called Cantor a "scientific charlatan," a "renegade," and a "corrupter of youth." The MacTutor biography documents Kronecker's hostility: Cantor stopped submitting to a major journal after clashes, and Kronecker never accepted an invitation to speak to the German mathematical society. Later, Ludwig Wittgenstein described set theory as "utter nonsense."

The personal toll is real but often misdescribed. Cantor's first recorded depressive episode came at the end of May 1884, and sanatorium stays began from 1899. MacTutor stresses that his depression was not caused by his mathematical worries, though it magnified them. He died of a heart attack on January 6, 1918, in Halle, in a sanatorium where he had spent his last year.

In the end the mathematics won. David Hilbert declared in 1926 that "no one shall expel us from the paradise that Cantor has created." Set theory is now the foundation under almost everything else in mathematics. The wider story of how mathematical ideas rise and fall is told in the history of mathematics from Euclid to Turing.

Do physicists use infinity?

Physicists run into infinity differently. In quantum electrodynamics, calculations of an electron's interactions involve adding up contributions from virtual particles at every possible energy, and the sums often diverge to infinity. Wikipedia's article on renormalization says the problem was first met by Born, Heisenberg, Jordan, and Dirac in the 1930s and tamed in 1947 to 1949 by Kramers, Bethe, Schwinger, Feynman, and Tomonaga, with Freeman Dyson systematizing it in 1949. The trick is to rewrite the "bare" mass and charge in terms of measured values and cancel the infinite leftovers.

Not everyone was comfortable. In 1975, Dirac objected that neglecting infinities is "just not sensible mathematics," and Feynman called the method a "dippy process." The modern view, developed by Kenneth Wilson (Nobel Prize in Physics, 1982), treats these theories as effective descriptions that hold up to a certain energy scale. An infinity in the equations is read as a warning that the theory has run past its range, which is one reason quantum gravity remains so hard.

Cosmology has its own version. Some models imply an infinite universe, but we can only ever observe a finite patch, about 93 billion light-years across. The number of galaxies in it is large and finite, as covered in how many galaxies are in the universe. Whether anything real is actually infinite is not settled.

Why bigger numbers matter for computing and codes

Large numbers are not only recreation. Modern encryption depends on numbers hundreds of digits long, and the security of schemes described in how encryption works rests on the fact that some problems on such numbers are easy to set up and brutally hard to reverse. The study of how fast functions grow, which Busy Beaver and TREE push to the limit, is also a study of what computers can and cannot do, since the Busy Beaver's growth is tied directly to the halting problem. Finite programs can be listed, so there are only countably many of them, while the real numbers are uncountable. That mismatch means most real numbers cannot be produced by any program at all.

The takeaway

There is no biggest number, and the reason is almost embarrassingly simple. The interesting part is what mathematicians did with that fact. They built notations that name numbers no physical universe could store, found that the biggest of them are measures of computation rather than of counting, and then crossed into infinity, where Cantor showed with a single diagonal argument that some sets simply cannot be listed.

The childhood game gets a respectable answer. "A trillion trillion" loses to a googol, a googol loses to a googolplex, and all of those lose to Graham's number and TREE(3). Infinity then splits: a countable infinity can be walked through step by step, and an uncountable one cannot. The lesson running through all of it is that the size of a number is less about its digits than about the rule that defines it.

FAQ

Q: What is the biggest number? A: There is no biggest number, since adding one to any whole number gives a larger one. The largest numbers with precise, standard definitions include Graham's number and TREE(3), and functions such as Busy Beaver grow faster than any computable function. Each is finite, and each can still be exceeded.

Q: Is infinity + 1 bigger than infinity? A: For cardinal numbers, which measure size, ℵ₀ + 1 = ℵ₀, so adding one changes nothing. For ordinal numbers, which record order, ω + 1 is a different number from ω, but the set it describes still has size ℵ₀. In ordinary calculus, infinity is a limit and not a number you can add to.

Q: What is the difference between countable and uncountable infinity? A: A countably infinite set can be listed, so each element gets a natural-number position (first, second, third). The integers and the fractions are countable. An uncountable set, such as the real numbers, cannot be listed, because a diagonal argument always produces an element missing from any proposed list.

Q: Is a googolplex bigger than the number of particles in the universe? A: Far bigger. Estimates for the observable universe run to about 10^80 atoms, and even packing it with neutrons would give about 10^128. A googolplex is 10^(10^100), so even writing its digits would need more places than there are atoms to hold them.

Q: What is the smallest infinity? A: ℵ₀ (aleph-null), the size of the counting numbers. A set has this size when its elements can be matched one-to-one with the counting numbers. The integers and the rational numbers are the same size, which surprises most people the first time.

Q: Is TREE(3) really larger than Graham's number? A: Yes. TREE(1) = 1 and TREE(2) = 3, but TREE(3) is so large that Graham's number is surpassed by comparison, according to the standard reference accounts. Both numbers are finite, and both are far too large to write out.

Q: Why can't we settle the continuum hypothesis? A: Gödel showed in 1938 that it cannot be disproved from the standard axioms, and Cohen showed in 1963 that it cannot be proved from them. Settling it would require adopting new axioms, and mathematicians disagree about which, if any, are justified.

Sources

• Scott Aaronson, "Who Can Name the Bigger Number?"
• MacTutor: Georg Cantor
• Stanford Encyclopedia of Philosophy: The Continuum Hypothesis
• Wikipedia: Graham's number
• Wikipedia: TREE(3)
• Wikipedia: Busy beaver
• Wikipedia: Skewes's number
• Wikipedia: Cantor's diagonal argument
• Wikipedia: Renormalization

Related Reading

• What Is the Largest Known Prime Number — And Why Do Mathematicians Keep Searching?
• What Is the Higgs Boson — and Why It Matters
• What Is Entropy — and Why the Universe Keeps Running Down
• What Is Quantum Gravity — The Theory That Could Unite Physics?
• What Is the Speed of Dark — And Other Things That Move Faster Than Light