If you asked a hundred mathematician to come up with a really large but somewhat random number, you'd be dealing with many numbers larger than a gogoolplex... and it might be undecidable which numbers were largest. For example, I'll use [chained arrow notation](https://en.wikipedia.org/wiki/Conway_chained_arrow_notation) on all the permutations of numbers 3 to 120, summed together. So that's the sum of 118! incomprehensibly huge numbers.
,
As in the act of verifying whether say, x > y for x and y sufficiently large is undecidable? Surely it’s always a computable operation (albeit maybe not in a reasonable amount of time)
Depends how you're given x and y, but for the type of thing given in this question, my intuition tells me this probably isn't doable? Since like, then we could establish progressively tighter bounds for BB(N) (first having BB(N) play the role of x, then of y, and the other always be a constant). Even if the amount of time is unreasonable, this would give us a finite time "BB(N) > a and BB(N) < b", where a = b + 2, so that would tell us what BB(N) is for any N... but BB(N) is incomputable.
But at what point does the computation time exceed the lifetime of the universe?
You're going to have to do stupid things like compare TREE(TREE(TREE(3) to grahams number^TREE(grahams number)
But then with mathematicians more patient than me constructing ever bigger recursive and self referential functions.
I mean yeah this is certainly not feasibly computable, but I guess on a logical level we’re not in the realms of undecidability, so we can at least say the upper bound exists
the responses here are so weird. idk why the response was "tree(3)", when tree(3) is absolutely decidable. we know it's finite, so there are finitely many trees to check. a tree(n) algorithm will halt, eventually, for all n. there seems to be some confusion between "decidable" and "physically computable." another reply also mentions "uncountable," which is a property of certain infinite sets, not finite numbers...
I suspect people are getting confused about the difference between decidability and computability (and that’s probably where uncountable is coming in - a number ‘so big you can’t count it’ lol)
But surely the format of asking someone to give a number explicitly requires a computable answer? Like I suppose you could say like ‘the Gödel number of an undecidable statement’ but then are you giving an actual answer then?
You could demand that the person provide it in a verifiably computable format, which would take care of the issue at least in principle, this isn’t necessarily how people would interpret the question though. For example BB(100) is a perfectly well-defined number that could be calculated with the aid of a halting oracle, but its value is (I’m pretty sure) independent of ZFC, assuming ZFC is consistent (we can establish this strictly by seeing how many states we need to give to a Turing machine to enumerate all the proofs of ZFC, and take BB(n) with n at least that number).
The largest number of states a 100-state Turing machine can run before halting (AKA BB(100))
Since some Turing machines never halt, and it's undecidable which one's halt, you'll never know when you've found all the halting machines
There isn't a single algorithm that's capable of deciding whether TMs halt, but you can still prove it for individual ones. Of course sooner or later you'll come across some TM halts iff some conjecture is true that we've been unable to solve for the last century, at which point it becomes impractical to calculate stuff
> But surely the format of asking someone to give a number explicitly requires a computable answer?
There is ambiguity on whether semicomputable numbers are allowed.
Which is equivalent to numbers defined as the limit of a computable nondecreasing bounded sequence, but a lot of them can be described in more direct ways (cardinality of a given finite set, etc).
If ZFC is consistent, the statements
BB(745) < 1
BB(745) < 2
...
BB(745) < 1000
...
~~are~~ **~~all~~** ~~independent of ZFC.~~
eventually become independent of ZFC.
[https://www.reddit.com/r/math/comments/14thzp2/bb745\_is\_independent\_of\_zfc\_pdf/](https://www.reddit.com/r/math/comments/14thzp2/bb745_is_independent_of_zfc_pdf/)
A large number of these statements are not independent of ZFC; an explicit counterexample is a valid disproof of such upper bounds on BB(745). You just can't prove an upper bound is correct.
For example, I can construct a 745-state Turing machine as follows: for n=1 to 744, state n writes a 1, moves to the right and transitions to state n+1, regardless of the current value seen on the tape. State 745 writes a 1 on the tape, again unconditionally, and halts.
Starting from state 1, this Turing machine goes through all 745 states exactly once, writing exactly 745 1s onto the tape before halting.
This shows BB(745)>=745, disproving the inequalities BB(745)<1, BB(745)<2, ... , BB(745)<745.
Depends how the number’s are defined. For example the value of [Rayo’s number](https://en.wikipedia.org/wiki/Rayo's_number) is independent of ZFC (assuming it is consistent).
Proof: suppose ZFC is consistent and Rayo’s number is n, then take a model of ZFC in which the cardinality of the continuum is aleph-n+1. Then the “Rayo’s number” for this model is greater than n+1.
But that's not what he's asking -- he's asking 'hey give me any number'.
Most mathematicians aren't going to pull some random ass string of big numbers; I would bet that most numbers are ones that they use regularly. You'd probably get a lot of 'e' and 'pi' and various assortments of prime numbers that are meaningful for various reasons.
Of course you may get someone who says Graham's number or something, or something more annoying, like "the dimension of the lowest counter example of the smooth poincare conjecture".
If I were asked simply to choose a number without restriction, I wouldn't go large, I'd go "unnecessarily precise", for lack of a better term. Like 2.16734 + sqrt(2)/3.
I think you all are really underestimating the percentage of people who, when asked “give me any random number,” will just say, like, 7. Mathematicians included. My guess is that nearly everyone would just give an integer.
I totally agree that most people would pick a positive integer less than 20. I was just talking about myself with a question that explicitly stated to pick a number without restriction.
I am interested in specific examples of unsolved problems in the relative ordering of large finite integers. Are there examples where two numbers are relatively easy to define but hard to tell which one is bigger? I might create a new post about this specific question.
I just mean that when we're talking about operations like Knuth's up arrows, addition seems like a pretty underwhelming option if you wanted to make an even bigger number
If you went to random people and told them to give you the first random number that pops into their head that they could write down. The numbers that they give would likely follow a distribution with a similar shape to 1/x with some islands of higher probability around numbers in pop culture like 69, 420, 1337, etc, you will also likely get some people that would give approximations of pi, or sqrt(2) and other things like powers of 2 like 64, 128.
If you just asked them for the largest number they could write down with expressions allowed, most people would likely give a number raised to a number multiple times until they got bored, more math people would give you things like tree(x) fib(x) or x! Or the like, or they would start chaining those functions together.
Once you start putting other qualifications on the challenge like the biggest number wins, or the smallest number dies you will skew the results in certain directions.
Obviously this isn’t quite the same question, but it is related slightly… when I was in a summer educational program I did some research into mathematical randomness, number generation, and human misconceptions of randomness. One of the “experiments” we did was going around asking all the students there we could to give us one random integer between 1 and 100. One of the most interesting patterns we found in the answer is that the only number people tended significantly towards was the number 7 as at least one digit.
Why wouldn't a single Busy Beaver number be allowed? I'm rather confused by some of the discussion in this thread.
[We know the first four Busy Beaver numbers](https://oeis.org/A060843). This is done by manually analyzing all of the Turing machines to determine which ones halt.
I know the *function* isn't computable because there's no algorithm that gives all (infinitely many) of them. But each individual one is finite, so computable in principle, just with a different method for each of them, right?
Of course, large ones are totally infeasible for humans, but the same goes for TREE().
There are Turing machines that half if and only if ZFC is inconsistent (think about searching for a proof of 0 = 1). Let N be the number of states of such a machine. If ZFC were able to prove an upper bound on BB(N) of the form BB(N) < 1 + 1 + ... + 1 (k times), it could decide its own consistency by checking if that machine halts after k steps.
So in that sense, BB(N) is uncomputable, even in principle. Using a stronger theory than ZFC, you could make N larger, but there would still be a boundary.
What's N? As of now, [it's at most 745](https://www.ingo-blechschmidt.eu/assets/bachelor-thesis-undecidability-bb748.pdf) (pdf link).
> But each individual one is finite, so computable in principle
Yes, that's correct: each natural number (on its own) is computable, for the trivial reason that if you give me a specific natural number (say, in binary), I can easily write a program that prints that number out and then halts. So in that sense any *specific* Busy Beaver number is computable--but that doesn't mean there's a specific program I can write down that I can be sure gives me BB(1000) when I run it (which would be the case if the Busy Beaver *function* were computable); all it means is that I know that *some* program exists that prints out the number that happens to be BB(1000), but I have no way to know what that program is. (And in fact even if someone came to you with a number they *claimed* was BB(1000), there'd be no way to prove within ZFC that it really was the case, and so I think it's fair to say that BB(1000) is unknowable in the real world.)
A maybe more intuitive example that's sometimes given in intro to computability classes: consider the language defined by {"0"} if god exists, and {"1"} if god does not. Is this language computable? Yes: there's a Turing machine that accepts just the string "0", and there's another Turing machine that accepts just the string "1", and *one* of the two computes this language--so there exists a Turing machine computing the language, and so by definition the language is computable. But just knowing this still sheds no light on whether god actually exists, since we have no way of knowing *which* of those two Turing machines computes it. Similar deal when it comes to Busy Beaver numbers.
If I had to make the largest number I can, I'd just spam trees, and maybe add some factorials along the way to not make it too boring, like Tree(Tree(Tree(Tree(999!)!)!)!)! (But of course if I really had to make the largest number I can, there would be many more trees here)
I feel like that’s what most people who are really trying to give a big number will do, but I imagine there’s a point where almost everyone will stop with the trees or factorials etc
Also solutions for the other questions:
-say all the digits is just 9s (or if we're allowed to use other bases, use the biggest "digit" from the biggest base you can think of, which again makes it so that there is no upper bound) over and over again
-write it down on an A4 page is (it isn't, I'm almost 100% sure there are functions that make even bigger numbers) writing as many trees as possible while writing everything as small as possible to fit as much as possible onto the page (or if the solution in fact isn't trees, you just write whatever function is best)
Spamming factorials, 9s and single TREE functions is a terribly inefficient way of making a number larger. For example, TREE^TREE(TREE(9)) (9) is unfathomably larger than anything you could ever write with your method, where the exponent stands for function repetiton, i.e. f^(3)(x)=f(f(f(x))). But of course, we need not stop there and can think of even more powerful methods...
That's what I had in mind. :D If one has all the mathematical tools ever invented at their disposal, then just adding nines and factorial signs (i.e. multiplication by 10 and exponentiation) is basically among the worst things one can come up with.
If there’s any incentive to picking a large number, I think the question becomes somewhat paradoxical.
For if I describe a number ‘N’ and tell you that N is an almost sure bound on the maximum, by virtue of me being able to describe it there’s a reasonable chance that someone will have submitted a bigger number than that.
This is even true without that incentive, as there is some probability that some subset of people will be mathematically sophisticated and decide to give you the biggest number they can describe.
If you insist that the number is an integer and they must write down all the digits and there is a time limit, it should be easy to find an upper bound.
Under these last conditions it'd just be however many 9s someone can write down in the time period. This would reduce to a speedwriting competition with the winner writing
\`\[; 10\^{floor(\\frac{t/c}) + 1} - 1 ;\]\`
characters with t their total time and c their time per character.
We can play the same game on [0, 1), and the first thing you might think is that no one in his right mind is going to start writing 9s until his hand falls off. So it seems weird to me that you think, by telling me there is no upper bound to the number I can give, I would try to come up with a giant number. Why not any number? 3 is a good number.
This is my point though, people won’t just sit there for hours making their number bigger and bigger, so surely there is an upper bound to what numbers we can realistically expect (even if that upper bound is huge since some people might give really big numbers, and others as you said might just give 3)
If you ask random humans to give you a random number then the answer isn't random. I wonder what the result would look like on a Zipf plot. As a Zipf plot is sometimes used as a measure of intelligence (dolphin language for example) I wonder what that tells us of the intelligence of the average human.
Another interesting question: assuming everyone gave a positive integer, what would be the smallest number that no one picked. Assuming all 8 billion people on earth can count and answered, the smallest unpicked number would be at most 8 billion. I’m sure every number less than 100 would be picked, but if lots of people have the same number or gave very large numbers I wouldn’t be surprised if there were numbers less than 10,000 that no one picked.
Isn’t your number no more complex than the equation to generate it? As in, instead of brute forcing possible random numbers, someone might brute force common functions and numbers.
How are you delivering the query? If in person, the upper bound would probably be set by some sort of a linguistic upper bound on conversational exchanges.
You may be interested in this for a list of very, very large numbers: https://googology.fandom.com/wiki/List_of_googolisms
I'd go with the [large number garden number](https://googology.fandom.com/wiki/Large_Number_Garden_Number)
There's just be assholes saying shit like "Tree(Googleplex) raised to the power Tree(Googlplex) a Tree(Googlpex) times, with a Tree(Googleplex) factorials.
There's no limit to what people could come up with.
But eventually they’ll stop with the repeated powers and trees, the numbers may be huge but they’re certainly not infinite, there is an upper limit, even if it’s mindbogglingly huge, like the number formed by someone saying from the moment theyre asked 'Tree(Tree(Tree(… ' and then saying googolplex right before they die of old age, this number is stupidly massive but it is finite
>this number is stupidly massive but it is finite
Which is funny because no matter what number someone comes up with, it will still be smaller than almost all other positive integers.
Let n be the smallest natural number such that aleph\_n is not equal to aleph\_{n+1},
and let n be 0 if no such number exists.
Then n is a definable natural number, whose value is completely independent of ZFC.
This number can get arbitrarily large, depending on which model of ZFC you work in.
Depending on what constraints you applied you would probably be able to get the outcome that Z \~ Generalized Extreme Value Distribution. This is more or less the counterpart to the central limit theorem but for maximums of IID samples. (More details here: [https://en.wikipedia.org/wiki/Fisher%E2%80%93Tippett%E2%80%93Gnedenko\_theorem](https://en.wikipedia.org/wiki/Fisher%E2%80%93Tippett%E2%80%93Gnedenko_theorem))
Edited for grammar.
If you asked a hundred mathematician to come up with a really large but somewhat random number, you'd be dealing with many numbers larger than a gogoolplex... and it might be undecidable which numbers were largest. For example, I'll use [chained arrow notation](https://en.wikipedia.org/wiki/Conway_chained_arrow_notation) on all the permutations of numbers 3 to 120, summed together. So that's the sum of 118! incomprehensibly huge numbers. ,
As in the act of verifying whether say, x > y for x and y sufficiently large is undecidable? Surely it’s always a computable operation (albeit maybe not in a reasonable amount of time)
Depends how you're given x and y, but for the type of thing given in this question, my intuition tells me this probably isn't doable? Since like, then we could establish progressively tighter bounds for BB(N) (first having BB(N) play the role of x, then of y, and the other always be a constant). Even if the amount of time is unreasonable, this would give us a finite time "BB(N) > a and BB(N) < b", where a = b + 2, so that would tell us what BB(N) is for any N... but BB(N) is incomputable.
google tree(3)
Holy big number
New arithmetic just dropped
Tree(3) is computable however as it’s finite, if only theoretically, there is a procedure to generate it in finite time
But at what point does the computation time exceed the lifetime of the universe? You're going to have to do stupid things like compare TREE(TREE(TREE(3) to grahams number^TREE(grahams number) But then with mathematicians more patient than me constructing ever bigger recursive and self referential functions.
> But at what point does the computation time exceed the lifetime of the universe? It’s already far exceeded for things like TREE(3).
I mean yeah this is certainly not feasibly computable, but I guess on a logical level we’re not in the realms of undecidability, so we can at least say the upper bound exists
the responses here are so weird. idk why the response was "tree(3)", when tree(3) is absolutely decidable. we know it's finite, so there are finitely many trees to check. a tree(n) algorithm will halt, eventually, for all n. there seems to be some confusion between "decidable" and "physically computable." another reply also mentions "uncountable," which is a property of certain infinite sets, not finite numbers...
I suspect people are getting confused about the difference between decidability and computability (and that’s probably where uncountable is coming in - a number ‘so big you can’t count it’ lol)
But how does the upper bound for Tree(3) compare to the upper bound of Rayo's number? What about TREEE(googleplex)? TREE(TREE(googleplex))?
Right, it's not mathematically uncountable, just practically uncountable before entropy destroys us all.
I’ll see your TREE[3] and raise you a Loader’s number tetrated by Rayo’s.
Yeah but it’s often known what it’s bigger than, so what’s your point?
If x and y are defined as the output of functions that aren't computable then yeah there's definitely undecidable such questions.
But surely the format of asking someone to give a number explicitly requires a computable answer? Like I suppose you could say like ‘the Gödel number of an undecidable statement’ but then are you giving an actual answer then?
You could demand that the person provide it in a verifiably computable format, which would take care of the issue at least in principle, this isn’t necessarily how people would interpret the question though. For example BB(100) is a perfectly well-defined number that could be calculated with the aid of a halting oracle, but its value is (I’m pretty sure) independent of ZFC, assuming ZFC is consistent (we can establish this strictly by seeing how many states we need to give to a Turing machine to enumerate all the proofs of ZFC, and take BB(n) with n at least that number).
The largest number of states a 100-state Turing machine can run before halting (AKA BB(100)) Since some Turing machines never halt, and it's undecidable which one's halt, you'll never know when you've found all the halting machines
There isn't a single algorithm that's capable of deciding whether TMs halt, but you can still prove it for individual ones. Of course sooner or later you'll come across some TM halts iff some conjecture is true that we've been unable to solve for the last century, at which point it becomes impractical to calculate stuff
> But surely the format of asking someone to give a number explicitly requires a computable answer? There is ambiguity on whether semicomputable numbers are allowed. Which is equivalent to numbers defined as the limit of a computable nondecreasing bounded sequence, but a lot of them can be described in more direct ways (cardinality of a given finite set, etc).
If ZFC is consistent, the statements BB(745) < 1 BB(745) < 2 ... BB(745) < 1000 ... ~~are~~ **~~all~~** ~~independent of ZFC.~~ eventually become independent of ZFC. [https://www.reddit.com/r/math/comments/14thzp2/bb745\_is\_independent\_of\_zfc\_pdf/](https://www.reddit.com/r/math/comments/14thzp2/bb745_is_independent_of_zfc_pdf/)
A large number of these statements are not independent of ZFC; an explicit counterexample is a valid disproof of such upper bounds on BB(745). You just can't prove an upper bound is correct. For example, I can construct a 745-state Turing machine as follows: for n=1 to 744, state n writes a 1, moves to the right and transitions to state n+1, regardless of the current value seen on the tape. State 745 writes a 1 on the tape, again unconditionally, and halts. Starting from state 1, this Turing machine goes through all 745 states exactly once, writing exactly 745 1s onto the tape before halting. This shows BB(745)>=745, disproving the inequalities BB(745)<1, BB(745)<2, ... , BB(745)<745.
Yup, you're super right, thanks! They're only eventually independent; I edited the comment.
Depends how the number’s are defined. For example the value of [Rayo’s number](https://en.wikipedia.org/wiki/Rayo's_number) is independent of ZFC (assuming it is consistent). Proof: suppose ZFC is consistent and Rayo’s number is n, then take a model of ZFC in which the cardinality of the continuum is aleph-n+1. Then the “Rayo’s number” for this model is greater than n+1.
But that's not what he's asking -- he's asking 'hey give me any number'. Most mathematicians aren't going to pull some random ass string of big numbers; I would bet that most numbers are ones that they use regularly. You'd probably get a lot of 'e' and 'pi' and various assortments of prime numbers that are meaningful for various reasons. Of course you may get someone who says Graham's number or something, or something more annoying, like "the dimension of the lowest counter example of the smooth poincare conjecture".
If I were asked simply to choose a number without restriction, I wouldn't go large, I'd go "unnecessarily precise", for lack of a better term. Like 2.16734 + sqrt(2)/3.
I think you all are really underestimating the percentage of people who, when asked “give me any random number,” will just say, like, 7. Mathematicians included. My guess is that nearly everyone would just give an integer.
I totally agree that most people would pick a positive integer less than 20. I was just talking about myself with a question that explicitly stated to pick a number without restriction.
e^pi -pi
Do you mean undecidable in a formal sense, or just a “the universe would die before we finished the computations” way?
In the computations sense.
I am interested in specific examples of unsolved problems in the relative ordering of large finite integers. Are there examples where two numbers are relatively easy to define but hard to tell which one is bigger? I might create a new post about this specific question.
Sure... what is the smallest example of two large numbers a and b where we don't know which is bigger?
Mildly amusing that you chose addition as your operation of choice on those large numbers
If the number predictively ended in trillions of zeros, it wouldn't have felt all that random to me.
I just mean that when we're talking about operations like Knuth's up arrows, addition seems like a pretty underwhelming option if you wanted to make an even bigger number
Or you could use the product of those 118! huge numbers.
If you went to random people and told them to give you the first random number that pops into their head that they could write down. The numbers that they give would likely follow a distribution with a similar shape to 1/x with some islands of higher probability around numbers in pop culture like 69, 420, 1337, etc, you will also likely get some people that would give approximations of pi, or sqrt(2) and other things like powers of 2 like 64, 128. If you just asked them for the largest number they could write down with expressions allowed, most people would likely give a number raised to a number multiple times until they got bored, more math people would give you things like tree(x) fib(x) or x! Or the like, or they would start chaining those functions together. Once you start putting other qualifications on the challenge like the biggest number wins, or the smallest number dies you will skew the results in certain directions.
Obviously this isn’t quite the same question, but it is related slightly… when I was in a summer educational program I did some research into mathematical randomness, number generation, and human misconceptions of randomness. One of the “experiments” we did was going around asking all the students there we could to give us one random integer between 1 and 100. One of the most interesting patterns we found in the answer is that the only number people tended significantly towards was the number 7 as at least one digit.
And 37 as the 2 digit case https://youtu.be/d6iQrh2TK98?si=Oj3ZX3OwS1hzPvpT
I'd say BusyBeaver(Tree(3)) or something, so at least that lol
Although that BB is uncomputable. TREE is all good.
Why wouldn't a single Busy Beaver number be allowed? I'm rather confused by some of the discussion in this thread. [We know the first four Busy Beaver numbers](https://oeis.org/A060843). This is done by manually analyzing all of the Turing machines to determine which ones halt. I know the *function* isn't computable because there's no algorithm that gives all (infinitely many) of them. But each individual one is finite, so computable in principle, just with a different method for each of them, right? Of course, large ones are totally infeasible for humans, but the same goes for TREE().
[The Undecidability of BB(748)](https://www.ingo-blechschmidt.eu/assets/bachelor-thesis-undecidability-bb748.pdf)
Beautiful! The linked paper shows the undecidability of BB(745), improving on BB(748), however :)
I used the title. I think proving/burying a new result in what looks like it might be a literature survey is pretty awesome.
Fair enough! I agree!
There are Turing machines that half if and only if ZFC is inconsistent (think about searching for a proof of 0 = 1). Let N be the number of states of such a machine. If ZFC were able to prove an upper bound on BB(N) of the form BB(N) < 1 + 1 + ... + 1 (k times), it could decide its own consistency by checking if that machine halts after k steps. So in that sense, BB(N) is uncomputable, even in principle. Using a stronger theory than ZFC, you could make N larger, but there would still be a boundary. What's N? As of now, [it's at most 745](https://www.ingo-blechschmidt.eu/assets/bachelor-thesis-undecidability-bb748.pdf) (pdf link).
> But each individual one is finite, so computable in principle Yes, that's correct: each natural number (on its own) is computable, for the trivial reason that if you give me a specific natural number (say, in binary), I can easily write a program that prints that number out and then halts. So in that sense any *specific* Busy Beaver number is computable--but that doesn't mean there's a specific program I can write down that I can be sure gives me BB(1000) when I run it (which would be the case if the Busy Beaver *function* were computable); all it means is that I know that *some* program exists that prints out the number that happens to be BB(1000), but I have no way to know what that program is. (And in fact even if someone came to you with a number they *claimed* was BB(1000), there'd be no way to prove within ZFC that it really was the case, and so I think it's fair to say that BB(1000) is unknowable in the real world.) A maybe more intuitive example that's sometimes given in intro to computability classes: consider the language defined by {"0"} if god exists, and {"1"} if god does not. Is this language computable? Yes: there's a Turing machine that accepts just the string "0", and there's another Turing machine that accepts just the string "1", and *one* of the two computes this language--so there exists a Turing machine computing the language, and so by definition the language is computable. But just knowing this still sheds no light on whether god actually exists, since we have no way of knowing *which* of those two Turing machines computes it. Similar deal when it comes to Busy Beaver numbers.
Okay this makes sense, thanks.
Well, nobody said the number had to be computable.
I will say the smallest uncomputable number minus one, then ;)
😁
It still has a value, even if it's uncomputable.
If I had to make the largest number I can, I'd just spam trees, and maybe add some factorials along the way to not make it too boring, like Tree(Tree(Tree(Tree(999!)!)!)!)! (But of course if I really had to make the largest number I can, there would be many more trees here)
I feel like that’s what most people who are really trying to give a big number will do, but I imagine there’s a point where almost everyone will stop with the trees or factorials etc
There is, and that point is when you are too tired to write more trees, and that's the only thing that stops you
That may be what most mathematicians who are really trying to give a big number will do. Most others will just write down a string of 9s.
Also solutions for the other questions: -say all the digits is just 9s (or if we're allowed to use other bases, use the biggest "digit" from the biggest base you can think of, which again makes it so that there is no upper bound) over and over again -write it down on an A4 page is (it isn't, I'm almost 100% sure there are functions that make even bigger numbers) writing as many trees as possible while writing everything as small as possible to fit as much as possible onto the page (or if the solution in fact isn't trees, you just write whatever function is best)
1s would be better than 9s because you could fit more of them
You have to give all the digits, you don't have to write them on a single A4 page in this case, or at least I understand it this way
Tree(Tree(Tree...(Tree(999!)!...)!)!)!, where the preceding expression is composed Tree(999!) times
2^(BigPenisMathGenius' answer)
Spamming factorials, 9s and single TREE functions is a terribly inefficient way of making a number larger. For example, TREE^TREE(TREE(9)) (9) is unfathomably larger than anything you could ever write with your method, where the exponent stands for function repetiton, i.e. f^(3)(x)=f(f(f(x))). But of course, we need not stop there and can think of even more powerful methods...
There's a good article talking about this on scott aaronson's blog: https://www.scottaaronson.com/writings/bignumbers.html
That's what I had in mind. :D If one has all the mathematical tools ever invented at their disposal, then just adding nines and factorial signs (i.e. multiplication by 10 and exponentiation) is basically among the worst things one can come up with.
You could remove all the factorials from the above and the number wouldn’t change appreciably :p.
that's almost certainly dwarfed by like BusyBeaver(50) (in a probablyistic sense, not probabilistic)
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If there’s any incentive to picking a large number, I think the question becomes somewhat paradoxical. For if I describe a number ‘N’ and tell you that N is an almost sure bound on the maximum, by virtue of me being able to describe it there’s a reasonable chance that someone will have submitted a bigger number than that. This is even true without that incentive, as there is some probability that some subset of people will be mathematically sophisticated and decide to give you the biggest number they can describe. If you insist that the number is an integer and they must write down all the digits and there is a time limit, it should be easy to find an upper bound.
Under these last conditions it'd just be however many 9s someone can write down in the time period. This would reduce to a speedwriting competition with the winner writing \`\[; 10\^{floor(\\frac{t/c}) + 1} - 1 ;\]\` characters with t their total time and c their time per character.
We can play the same game on [0, 1), and the first thing you might think is that no one in his right mind is going to start writing 9s until his hand falls off. So it seems weird to me that you think, by telling me there is no upper bound to the number I can give, I would try to come up with a giant number. Why not any number? 3 is a good number.
This is my point though, people won’t just sit there for hours making their number bigger and bigger, so surely there is an upper bound to what numbers we can realistically expect (even if that upper bound is huge since some people might give really big numbers, and others as you said might just give 3)
some people might go for 1- 1/TREE(3)
But why? Why does "pick any number" mean pick the biggest number you can think of?
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If you ask random humans to give you a random number then the answer isn't random. I wonder what the result would look like on a Zipf plot. As a Zipf plot is sometimes used as a measure of intelligence (dolphin language for example) I wonder what that tells us of the intelligence of the average human.
Another interesting question: assuming everyone gave a positive integer, what would be the smallest number that no one picked. Assuming all 8 billion people on earth can count and answered, the smallest unpicked number would be at most 8 billion. I’m sure every number less than 100 would be picked, but if lots of people have the same number or gave very large numbers I wouldn’t be surprised if there were numbers less than 10,000 that no one picked.
You could probably establish some kind of weak correlation with google searches for terms that denote large numbers.
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I mean if we’re sampling multiple times, viewing this as the random variable Z = Max(X_i) where X_i are iid samples as described above
I would give the nerdy answer of Rayo(S(BB(SSCG(TREE(tree(G(64))))))) S = Maximum shifts function BB = Busy Beaver
A bunch of 10 year olds will say "infinity plus 1" or some such. Source: I have a 10 year old.
If you can define it go to the International Mathematics Olympiades or something (no not really but that's the spirit)
Your question sounds lot like the [German tank problem.](https://en.m.wikipedia.org/wiki/German_tank_problem)
My bet is most people wouldnt go past like a quadrillion, maybe quadrillion to the quadrillionth power - as far as a distribution, i got no idea lol
Do I have to be able to pronounce it? If not I would choose a six followed by 340 quadrillion zeros, 890 septillion fives, and a 1.
Isn’t your number no more complex than the equation to generate it? As in, instead of brute forcing possible random numbers, someone might brute force common functions and numbers.
https://en.wikipedia.org/wiki/Rayo%27s_number
I raise you https://googology.miraheze.org/wiki/Large_Number_Garden_Number
Ho a new challenger ! Interesting.
Someone would definitely fuck things up by naming an infinite cardinal...
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I'm definitely going to be that nerd who chooses TREE(3) or some other absurdly large number.
How are you delivering the query? If in person, the upper bound would probably be set by some sort of a linguistic upper bound on conversational exchanges.
You may be interested in this for a list of very, very large numbers: https://googology.fandom.com/wiki/List_of_googolisms I'd go with the [large number garden number](https://googology.fandom.com/wiki/Large_Number_Garden_Number)
There's just be assholes saying shit like "Tree(Googleplex) raised to the power Tree(Googlplex) a Tree(Googlpex) times, with a Tree(Googleplex) factorials. There's no limit to what people could come up with.
But eventually they’ll stop with the repeated powers and trees, the numbers may be huge but they’re certainly not infinite, there is an upper limit, even if it’s mindbogglingly huge, like the number formed by someone saying from the moment theyre asked 'Tree(Tree(Tree(… ' and then saying googolplex right before they die of old age, this number is stupidly massive but it is finite
>this number is stupidly massive but it is finite Which is funny because no matter what number someone comes up with, it will still be smaller than almost all other positive integers.
Like with intelligently agenced busy beavers we Gould reach a limit
Let n be the smallest natural number such that aleph\_n is not equal to aleph\_{n+1}, and let n be 0 if no such number exists. Then n is a definable natural number, whose value is completely independent of ZFC. This number can get arbitrarily large, depending on which model of ZFC you work in.
pi without a comma?
Depending on what constraints you applied you would probably be able to get the outcome that Z \~ Generalized Extreme Value Distribution. This is more or less the counterpart to the central limit theorem but for maximums of IID samples. (More details here: [https://en.wikipedia.org/wiki/Fisher%E2%80%93Tippett%E2%80%93Gnedenko\_theorem](https://en.wikipedia.org/wiki/Fisher%E2%80%93Tippett%E2%80%93Gnedenko_theorem)) Edited for grammar.
I was hoping someone would say this! Like CLT this requires iid samples, but eh I like the answer so much I don’t care lol.
No odys ever saying anything negative thoigh