In a September Habr translation, Joel David Hamkins dissects a toy-looking contest: what's the maximum number you can specify in a single 280-character tweet? Fill every digit with nines—you get 10²⁸⁰−1. Build factorial towers—you fly further. Then it turns out judging such a contest is impossible without solving the halting problem, and the article's closing line sounds like the manifesto of an entire underground sport: "definability is not definable from within the language itself."
I was hooked by how this race ended. The humanities discipline of "googology"—a competition in naming large numbers—ended in a triple dead end: a philosophical dispute over whether a champion even exists, a machine nobody can understand, and a wall of ignorance that over ten years slid from eight thousand states down to four hundred. Below is how we got there.
In the third century BCE, Archimedes wrote "The Sand Reckoner" for the king of Syracuse—a popular treatise against people who considered sand uncountable. Taking a myriad (ten thousand) as his base and Aristarchus' cosmology, he estimated the number of sand grains filling the universe at 10⁶³. For two thousand years this result held the bar. By Scott Aaronson's ironic estimate, 10⁶³ is the largest number with a common American name: vigintillion.
The next paradigm shift happened only in the twentieth century, and its harbinger remains the legend of the vizier who invented chess: one grain on the first square, two on the second, doubling to the sixty-fourth. The naive king promised a reward, not noticing the sum equals 2⁶⁴−1—about 18.4 quintillion grains; by the estimate in Aaronson's 1999 essay, that's world wheat production for 150 years.
In 1928 Wilhelm Ackermann broke the very concept of "growth ceiling": he constructed a computable function that outpaces any "primitive recursive" one. Behind this lies a ladder of operations—addition, multiplication, exponentiation, tetration, pentation and further up the steps—which Donald Knuth in 1976 packaged into arrow notation.
A genre classic is Graham's number. In 1977 Ronald Graham passed to Martin Gardner for Scientific American an estimate G = f⁶⁴(4), where f(n) = 3↑ⁿ3—three with n arrows to three. In 1980 the number made it into Guinness as the largest used in a published mathematical proof. It has so many digits that the observable universe can't contain even their count. Yet the last ten digits are known exactly: …2464195387—the number is a power of three, and the tail is computable by algorithm. Now the irony: Graham's problem itself shrank long ago. The best known bounds today are 13 ≤ N* ≤ N''. The record holder turned out to be an upper estimate for a problem whose answer might be thirteen.
The next step grew from Turing's definition of "computable." In 1962 Tibor Radó published in Bell System Technical Journal "On Non-Computable Functions" and defined BB(n): the maximum steps a Turing machine with n states takes on a blank tape, if it halts at all. The first values are modest: BB(1)=1, BB(2)=6, BB(3)=21 (Lin and Radó, 1965), BB(4)=107 (Brady, 1983).
Then the unimaginable begins. The champion machine for five states was found by Heiner Marxen and Jürgen Buntrock in 1989: it halts after 47,176,870 steps. Proving nobody halts longer took another 35 years—the open project bbchallenge tackled this, enumerating all one hundred million five-state machines and analyzing each one. In 2024 participant mxdys published Coq-BB5, a machine-checked proof in Rocq, and the quest stretching from 1962 closed. Almost all non-halting machines were handled by five families of automatic "deciders"; thirteen stubborn ones required individual proofs.
BB(6) remains unsolved.
The best lower bound (June 2025, again mxdys)—greater than 2↑↑2↑↑2↑↑10: a tetration tower with another tetration tower in its exponent. For calibration: 2↑↑5 = 2⁶⁵⁵³⁶, a number with 19,729 digits. The pursuit of BB isn't from sporting passion: knowing exact BB(n) solves the halting problem for all machines with n states.
There exists a 27-state machine that enumerates even numbers and halts upon finding the first not representable as a sum of two primes. If we knew BB(27), Goldbach's conjecture would be closed—even if the computation itself took longer than the universe's lifetime. Busy Beaver values are a compressed catalog of all "final" mathematics: every Π₁-sentence is encoded in them.
The most famous match in this discipline took place January 26, 2007 at MIT, during Independent Activities Period: philosopher Agustín Rayo versus Princeton philosopher Adam Elga. Rules: alternating moves, whoever writes a valid number last wins; only finite numbers; ban on semantic vocabulary—words like "names," "refers to," "true of." The third point decides everything: without it, Elga would win with the first move, writing "the largest number Agustín named, plus one."
By Rayo's own notes, the game went like this. Rayo wrote thirty to forty ones. Elga took an eraser, erased everything but the first two ones, and built a factorial tower—just 11!! gives roughly 6×10²⁸⁶⁰⁷⁸¹⁷⁰.
Rayo answered with BB(10¹⁰⁰): the productivity of the most productive Turing machine with no more than a googol states. Then came a ladder of oracles—BB₁, BB₂, up to BB_θ(10¹⁰⁰), where θ is the first non-recursive ordinal. The winning move: "the smallest number larger than any finite number named by an expression in the first-order language of set theory of length up to a googol symbols." The word "named" is forbidden by the rules, so Rayo replaced it with a formalization through Sat—a second-order formula constructing a non-semantic surrogate of the concept "named." Thus Rayo's number was born.
Popular retellings botch this game badly. A fresh 2026 piece confidently reports that Elga's first move was writing "1," and gives "BB values" of 1, 4, 6, 13, 4098—these are actually values of Σ, Radó's second function, which counts ones on the final tape, not steps. The primary source—Rayo's own textbook—contradicts both details. The meme about the most mathematical of duels mutates in copying, like any other meme.
The number itself is more interesting. Rayo honestly writes: a philosopher who doesn't accept Platonism in set theory might not recognize it as defined—any recursively enumerable axiomatization of second-order language has non-isomorphic models, and in different models the champion might be different. Aaronson in his 2020 survey puts it harsher: "I personally refuse to consider an integer sequence well-defined if its values can depend on the truth of the axiom of choice or the continuum hypothesis." The big number race has a champion whose title is disputed at the philosophy-of-science level.
If BB grows faster than anything computable, can we at least know its values? Here Gödel meets Radó. For any recursively enumerable and arithmetically sound theory T, there exists a threshold n beyond which T cannot prove values of BB(n). The reason is simple: a machine with k states that enumerates proofs in T and halts upon finding a contradiction is easy to construct. By proving the value of BB(k), the theory would thereby prove its own consistency—which is forbidden by Gödel's second theorem.
How many values are actually available to us? For a long time the answer was "unknown: ten or a million?" In 2016 Adam Yedidia and Aaronson constructed an explicit 7918-state machine (in the final paper version—7910) that halts if and only if the theory ZF plus "stationary Ramsey property" is inconsistent. This required writing the Laconic language, an interpreter of roughly 4000 states, and "introspective encoding" that saves by a factor of five. Stefan O'Rear simplified the construction to 1919, then to 748 states—and removed dependence on Harvey Friedman's results: his machine directly searches for a contradiction in ZF.
Then came the conveyor: 745 (Riebel, undergraduate thesis, 2023), 643 and 636 (Ridenour, 2024), 588, 549 and 432 (Wade, July–August 2025). For Peano arithmetic the wall stands at 372 states (House, February 2026).
Over ten years the boundary of provable ignorance slid from eight thousand states to four hundred and change. No revolution occurred—engineers simply optimized the compilation of Gödel's theorem into machine states. Aaronson compares this to the origin-of-life problem: also "a question of constants," only instead of a self-replicating molecule—a self-unknowable program.
Along the way it turned out: Goldbach's conjecture packs into 27 states (username Code Golf Addict), the Riemann hypothesis—into 744 (Matiyasevich, O'Rear, Aaronson). The value is that nobody will ever know BB(744) within ZF. Honesty caveat: the record machines of recent years haven't undergone formal verification, and the pioneering construction relied on a Friedman result that still hasn't appeared in print.
The only front where battle rages right now—BB(6)—has hit a problem older than bbchallenge itself. On June 28, 2024, participant mxdys isolated from the "holdouts" a machine nicknamed Antihydra: a cryptid. Its behavior reduces to three lines: h = 8, c = 0; at each step c += 2 if h is even, otherwise c −= 1; then h += h//2. Halt occurs at c = −1, that is, when odd sequence members outnumber even ones by more than two to one.
Mathematically this is D(0) = 7, D(n+1) = ⌈3D(n)/2⌉, and the halting question is a question about the parity of members of this sequence. The same mathematics governs Mahler's 1968 Z-numbers problem: how do fractional parts of numbers ξ·(3/2)ⁿ behave. Sixty years of progress—zero. Probabilistic arguments say Antihydra "probviously" runs forever: it has no reason to halt. The community cast an actual trophy for whoever solves the problem.
As of September 2026, there are 855 holdouts in BB(6) up to equivalence, all simulated to 10¹³ steps, a partial proof slowly maturing in Rocq. Shawn Ligocki summarized the situation in his analysis title: BB(6) is hard.
Berry's paradox helps assemble the picture, which Bertrand Russell attributed to librarian G. G. Berry: the phrase "the largest number nameable in one thousand characters" names that number in eighty characters—the snake swallows its own tail. Tarski proved truth cannot be defined from within language, and in 2010 Aaronson asked on MathOverflow whether one can surpass Turing machines using set theory means.
Hamkins' answer: if ZFC is consistent, there exist models of it in which "the collection of definable natural numbers" is not a set and not even a class. The instrument measuring champions itself falls into a hole.
David Simmons summarized with a typology: biggest number contests come in three kinds—won by whoever has more paper; whoever believes in stronger axioms; and philosophical, which can't be judged at all. Rayo himself in his textbook closes the topic with a phrase: the hunt for numbers turned into a hunt for languages. And why large numbers frighten is explained by neuropsychologist Stanislas Dehaene (Science, 1999): the brain has two counting mechanisms—an approximate "number line," shared with animals, and exact symbolic counting, tied to language. BB(1000) doesn't fit in the number line—that's where evolution ends and science begins.
The final answer to the hook question came out drier than promised. The biggest number you can fit in a tweet is the Busy Beaver of a machine with ~280 states, but writing its value is impossible in principle, and from one hundred-plus states upward the answer is provably inaccessible to ZFC. The number would win any contest, but nobody could hand over the prize. But in the bbchallenge office lies a real trophy—for whoever proves Antihydra doesn't halt. The race for the biggest number in the world got stuck at six states and a little machine that, most likely, just walks along its tape forever.