Why can’t people submit serious large number entries instead of this nonsense.

Here’s a blog post about large numbers, written in the style of the content in my large numbers website. I couldn’t fit it in any section of that website, so I decided to post it in this blog instead.

Introduction: Hamkins’ Large Number Contest

One day, when I was doing some online research about large numbers on the Internet, I came across this large number contest. It is a large number contest held by Joel David Hamkins and Ruizhi Yang at a top-three Chinese university, where you submit the largest number you can write down on an index card—quite a typical large number prompt. The entries were submitted by 150 undergraduate students at the beginning of a talk held at the university.

The rules for the contest were as follows:

1. A submission entry consists of the description of a positive integer written on an ordinary index card, using common mathematical operations and notation or English words and phrases.
2. Write legibly, and use at most 100 symbols from the ordinary ASCII character set. Bring your submission to the talk.
3. Descriptions that fail to describe a number are disqualified.
4. The submission with the largest number wins.
5. The prize will be $1 million USD, divided by the winning number itself, rounded to the nearest cent, plus another small token prize.

with 99999, 10*(10*99)+5, and “the population of Shanghai at the moment” listed as examples of valid submissions.

The first two of these are indisputably valid submissions in any large number contest. The third, however, is a little iffy. Undoubtedly there is a number that denotes the current population of Shanghai, but it isn’t easy to precisely determine what number you’re talking about. It’s kind of a physical quantity in that it counts a value in the physical world (technically human world, but it’s about the same thing), and since it’s hard to precisely determine the exact values of physical quantities, you should stray from giving those as examples of huge numbers. Besides, they aren’t that great; almost all physical quantities would be easily topped by a googolplex, which isn’t that big in the scope of large numbers, and every one of them, under any stretch of the term “physical quantity”, would fall well under a decker (10 tetrated to 10 = 10^10^10^10^10^10^10^10^10^10), which isn’t that hard of a number to think up if you’re reasonably clever.

Hamkins goes on to talk about ways to create large numbers. He talks about the googol and googolplex, the plex bang stack hierarchy, and up-arrow notation, even devising an extension to up-arrow notation to make even bigger numbers.

He then talks about a potential way to trump all this an even bigger entry: entering something like:

The largest number describable on an index card according to the rules of this contest.

First of all, that’s cheating because it’s making an entry that necessarily would trump all others. Even if we allow entries like this (which Hamkins seems to), he himself notes that even if we allow that cheating entry, we could write:

The largest number describable on an index card according to the rules of this contest, plus one.

That fits snugly inside the 100-character limit (96 characters), and it’s even larger than the previous entry! But wait a minute—we just described a number describable on an index card that is larger than the “largest number describable on an index card”. This leads to a paradox called Berry’s paradox. In a nutshell, this means that the phrase “the largest number describable on an index card according to the rules of this contest” might not really be meaningful!

You might argue, couldn’t we devise a loophole around that? You could try—good luck doing so without leading to a paradox regardless. In any case, THAT IS CHEATING!!! A lot of people will be impressed by devising a really big well-defined number such as the famous Rayo’s number (once honored as the largest named number) from scratch, but nobody would blink an eye at “the largest number anyone can think of, plus one”.

But since Hamkins didn’t stop people from making cheating entries, that’s exactly what a lot of people did. Here I’ll review the eight entries that Hamkins selected to discuss.

The Eight Entries

Entry One: “Let’s write down something about pi!”

The first entry listed is as follows:

The maximum length of the number “π”, by now human can describe physically in 10 days. [sic]

There are a couple things bad about this entry. First off, it breaks the unwritten rule that we should avoid physical constants; although Hamkins clearly allows physical constants, it’s bad practice in googology to do so. Secondly, the wording is ambiguous. What does it mean for pi to have a maximum length that humans can describe physically in 10 days? The largest amount of digits known 10 days from now? The largest all humans can write in 10 days? The largest amount they can sound out in 10 days?

The largest amount of digits have been found is around 10 trillion as of 2015, according to Wikipedia’s article on pi. That’s not that big if you want to make a really big number; a number like

999999999999999999999999999999999999999999999999999999999999999999999999

which is exactly the kind of number a lot of people come up with in an attempt to make the biggest number you can is way bigger.

Hamkins offers another interpretation of what the submitter said: the number of digits of pi all seven billion people can sound out in 10 days. If you can say five digits a second and do it for 10 days straight, you would sound out 4 million digits. Multiply it by 7 billion people, and you get roughly 10¹⁶ (ten quadrillion). Still not that big.

Overall, this entry isn’t going to cut it. It’s not only ambiguous, but not that big by any interpretation. I would disqualify this entry since the description is too vague to definitely describe a certain number. This first entry is pretty poor, but the rules were a little lax.

Entry Two: “I’m so clever, I came up with infinity!!!”

The second entry listed is simply:

∞

I don’t need to say what’s bad about this. It’s so blatantly cheating. It isn’t even allowed, since the rules say you have to write a positive integer, which infinity isn’t.

Entry Three: “Screw serious large numbers, I’ll talk about how much I love my girlfriend!”

The third entry is a weird one:

(Be sure to let her know I don’t care about the 1m $)

My affection for my girlfriend which values at least ω.

This is obviously nonsense. It’s cool that you love your girlfriend so much, but how in the hell would you measure affection for your girlfriend?! This obviously wasn’t a serious entry, but one notable thing about it is that the submitter says that it “values at least ω”. ω, if you didn’t know, is the Greek letter omega, which is used in math as one way to denote infinity (as an ordinal to be specific). And infinite numbers are explicitly forbidden from the competition.

By the way, this entry arguably breaks the 100-character rule: the text in the card as a whole is over 100 characters. But if you only count the second line as the entry (the first is clearly just a remark), then it follows the rule. This doesn’t change that this entry is unarguably invalid given its usage of infinity.

Entry Four: “I know the world’s largest number, it’s Graham’s number!”

The fourth and most qualifiable entry is:

G = 3^^......^3 \
    3^^......^3 |
         :      | 64 layers
         :      |
     3^^....^3  |
      3^^^^3    /

Graham's Number:
G = g_64, where g_1 = 3^^^^3, g_n = 3^(n-1)3

This entry is just Graham’s number, an infamous large number that is commonly thought of as the “world’s largest number”, or more specifically, as the “largest number used in a mathematical proof”. Its size easily blows the layman’s mind, and it has additional credit as a number that has been used in serious mathematics. I should note that Graham’s number is nowhere near being the world’s largest number OR the largest used in serious mathematics. It’s in fact an alternate easier-to-explain version of a number that was used as a very high upper-bound to the solution of a simple Ramsey theory problem.

If you’re not a seasoned googologist, you can think of where Graham’s number falls in the world of massive numbers this way: with Graham’s number, sure, it’s big, but it’s small enough that you can understand the process for getting to a number that big. On the other hand, larger numbers like TREE(3) (the most famous number larger than Graham’s, which has also been used in serious mathematics), are way too massive for you to be able to understand how to get to numbers that big.

Now this card, unlike all others so far, does indeed define a well-defined number, but talk about uncreative! Graham’s number is one of the most famous large numbers, so it’s not at all original to just submit Graham’s number to a large number contest and call it a day. I would add the restriction that if you have unusual mathematical notations (stuff besides the digits 0-9, parentheses, +, -, *, /, exponents, factorials), you should be able to explain it by yourself within the character limit. That would make the contest A LOT more interesting. You wouldn’t be able to throw in Graham’s number without having a concise explanation of up-arrow notation and the G_n notation used to define Graham’s number based upon standard math notation. And with that, it isn’t hard to make something to surpass that by expanding on the G_n notation that is typically used to define Graham’s number. The extensions to up-arrow notation Hamkins devised can also take you way beyond Graham.

So at least this entry is a real well-defined number, but it’s a really dull entry, and one that I think shouldn’t be legitimate without an explanation of the notations used to define it.

Entry Five: “Largest number plus one!”

The fifth entry is as follows:

The largest number explicitly described by others plus one.

Another cheating entry! It’s just adding one to the largest entry submitted, which is ridiculously illegitimate. It could technically be legitimate if nobody else does an entry like this, but if someone else does that, then it leads to a paradox again. Hamkins notes exactly this, and describes this submission as “clever”, which I disagree with. A clever entry would be one with wit and originality, while doing things like “the largest number plus one” is not original at all.

If there are no other entries like this (which is not the case since “the largest number plus one” is so common), then this could potentially win, but since it’s cheating, I would disqualify it.

Entry Six: “Second largest number plus one!”

The sixth entry is similar to the fifth:

Second largest of collected positive numbers plus one.

This is very similar to the fifth entry, but with an attempted loophole around Berry’s paradox. If you can’t say “the largest number plus one” because that would then be the largest number, why not say “the second largest number plus one”? Surely that won’t lead to a paradox, right? Wrong.

Hamkins presumes that “collected positive numbers” means all the numbers submitted in the entry. The second largest of those would be the largest entry besides this one, so this would be by definition one more than the next biggest entry. Now this isn’t any different from the fifth entry in this competition; it’s no different from “the largest number plus one” if you think about it!

Entry Seven: “I’ll try to have it both ways!”

The seventh entry is an unusual one:

N =
1, if no other card is qualified
M+1, if some other cards are qualified where M is the largest number defined

Before I say anything, this one breaks the 100 character rule, at 111 characters. Hamkins doesn’t note this however.

Anyway, this one is trying to win either way: either “the largest submitted number plus 1” if there are other qualified cards (I’ve said before that this is both cheating and leads to a paradox), or just 1 if there aren’t any. Why 1? Because then, the prize would be 1,000,000 divided by 1 = 1 million dollars! Most submitters forgot about that rule entirely.

Under the second case (yet another “largest number plus 1”), the submitter would probably have Graham’s number plus 1. But then, what about the previous two submissions, which were the same idea? They would both be Graham’s number plus two—but wait, this would then be Graham’s number plus three! And so on. This is a totally invalid entry, needless to say.

Entry Eight: “I’m not even trying to be subtle when cheating!”

Finally, the eighth entry is as follows:

Assuming everyone is rational, then the number is explicitly defined and it has to be 1.
 ↓
The least number that makes me one of the members who wins the game.

First off, like the third one this one may or may not break the 100-character rule: the whole text is 100 characters, but if you consider the first line to just be a remark about the entry (which is what Hamkins interpreted it as), then it is OK with the rule.

In any case, this is YET ANOTHER cheating entry, making a number that will definitely win the game. The phrase “one of the members who wins the game” implies that the submitter believes that there will be ties in this game. It’s almost the same as saying “the largest number plus one”.

The submitter notes that if everyone is rational, then they will all submit the number 1 in order to win a million dollars. And how wrong that was! Just look at people submitting infinity, Graham’s number, and variants of “the largest number plus one”. There is only one other person (submitter of entry 7) who even considered submitting the number 1.

Reviewing the Entries

Hamkins doesn’t explicitly say whether the eight entries he listed are the only ones submitted. He says that he collected a number of interesting submissions which he’ll comment on, out of presumably 150 submissions (given that 150 undergraduate students went to the talk). This implies that the eight submissions listed above, all but two of which were cheating, were the most interesting. So what about the other 142 entries? Possibly they were other entries submitting infinity or variants of “the largest number plus one”, and it is likely that many entries were stuff like 9999999999999999999999999 (a naive way to make the largest number you can). None of them are larger than Graham’s number though, since Hamkins notes that Graham’s number is the largest entry whose value doesn’t depend on other entries. Additionally, Hamkins picked eight entries to showcase that he found interesting, implying that the other 142 aren’t as interesting in his opinion. It’s weird though, that of the eight entries he picked, all but two of them were cheating. He probably has a different idea of what entries are interesting than I do. Maybe he’s really fascinated with the idea of Berry’s paradox or something, which makes sense given that he goes at length to talk about it.

In any case, god damn those are some disappointing submissions. A large portion of them are just cheating, the largest one is apparently Graham’s number out of all the large numbers you can think up, and there probably aren’t any submissions that are much more interesting than the ambiguous one relating to pi and the one which is just Graham’s number. What’s worse is, those are submissions from undergraduate students at an exclusive Chinese university! How can these results be so disappointing??? Here I will speculate as to why that is.

Note that a lot of the entries used English words to try and make a number that will necessarily win the game. I think they did that because the game allowed—encouraged even—the usage of English words in your number. So a lot of people took advantage of that and likely thought that they were the only one smart enough to write down a variant of “the largest number plus one”.

I really think Hamkins should have prevented people from submitting entries like that if he wanted the submissions to be actually interesting. I doubt that would stop a lot of people, but it could still make the entries a little more interesting. Also he should have allowed only building upon standard mathematical notation, i.e. variables and the symbols 0 1 2 3 4 5 6 7 8 9 + – * / ( ) ^ !, and anything else must be explained in terms of those. I could define up-arrow notation compactly like so:

c = 1: a<c>b = a^b
b = 1: a<c>b = a
otherwise: a<c>b = a<c-1>(a<c>(b-1))

and submit the entry:

c = 1: a<c>b = a^b
b = 1: a<c>b = a
otherwise: a<c>b = a<c-1>(a<c>(b-1))
My number is 9<9<9<9^99>9>9>9.

and that’s exactly 100 characters long. It doesn’t top Graham’s number but it’s still a start. It’s pretty easy to define numbers like this, and they actually match up with an ideal set of googological rules. So in a nutshell, one of the reasons why the results are disappointing is because the rules are rather lax.

A second reason why I think the results are so disappointing is because people are inclined to take the easy way out. That’s just human nature, the way people work. People always want to do things in the easiest way they can, which is generally not detrimental to people, but rather vital for their survival. People started riding horses because they didn’t want to spend hours walking to the store across the city and weeks going from New York to Florida or whatever. Typewriters were made so people could write stuff faster than how they can by hand. Calculators were made so that people won’t need to actually think or write stuff down to perform mathematical calculations but have a machine do it for them. Lighter fluid was invented so that people could start fires to cook food with ease without having to rub rocks vigorously. The Internet exists so that people can find information right from their own computers by typing a few search terms into Google rather than driving to the library and scouring the encyclopedias until they find the entry on what they need to know about.

And if there’s an easy way to try and come up with a large number, it’s triumphantly calling out “the world’s largest number plus one”. But that isn’t legitimate. Large number contests are not a chore or a daily routine, but an intellectual challenge, where the easy way out isn’t the good thing to do. But that doesn’t stop people from doing just that, blind to the idea that it’s meant as a creative exercise in forging your own large number. Even when people come up with their own large numbers, walls of nines are all but ubiquitous. In his essay “Who Can Name the Bigger Number?“, Scott Aaronson describes a large number contest in which one contestant simply wrote as many nines as he could on the card in the time limit, and the other wrote a similar wall of nines, but followed by a superscript 999. The first one is not at all clever, while the second one at least makes use of exponentiation but still gravitates heavily towards the idea that walls of nines are the way to go. The 999 could be replaced with 9^{9^9}, an expression that takes up the same number of digits but instead of being a number you can count in 15 minutes, it’s a 369 million digit number, bigger than a googol multiplied by itself a million times. You could also make a huge power tower of nines which zooms way past the googolplex or googolduplex or anything like that. But most people aren’t this creative when it comes to devising large numbers. Aaronson remarks in his essay that the results of his large number competitions are never quite what he’d hope.

Conclusion

I think that we can gather from all this that the large number competition Hamkins held had results that are disappointing to googologists because (1) the rules were lax and almost encouraged things like the largest number plus one, and (2) a lot of contestants tried to take the easy way out, which is human nature. Maybe if he allowed only mathematical expressions it could be more interesting? A competition with seriously interesting results is Bignum Bakeoff, a competition that was similar to this but required people to submit C programs within 512 characters. Of the twenty entries, only five of them were programs that would qualify as attempts to automatically win; the other fifteen varied quite widely, and the winner is Loader’s number, one of the largest numbers known. Overall, if you want to hold an interesting large number contest, you’ll need quite the array of restrictions in order to stop at least some people from resorting to cheap tricks.