# No I Can't Even Is Just The Beginning

April 24, 2017 11:59 PM Subscribe

*Weirdly, thinking about Graham’s number has actually made me feel a little bit calmer about death...*On coming to grips with g_64.

Hm, did that article put the expansion of the googol in the googolplex section by mistake?

Edit: Whoops, rendering error in my browser. All good!

posted by rum-soaked space hobo at 12:23 AM on April 25, 2017

Edit: Whoops, rendering error in my browser. All good!

posted by rum-soaked space hobo at 12:23 AM on April 25, 2017

I dunno, i don't think you get any higher than 24. Let it go.

posted by flod at 1:10 AM on April 25, 2017 [17 favorites]

posted by flod at 1:10 AM on April 25, 2017 [17 favorites]

I’d already lost my grip before he got to ‘We’re about to enter a whole new realm of craziness, and I’m gonna say some shit that’s not okay. Are you ready?’ and, while I wasn’t really ready, I very much enjoyed the piece: thanks for posting it!

posted by misteraitch at 2:06 AM on April 25, 2017 [5 favorites]

posted by misteraitch at 2:06 AM on April 25, 2017 [5 favorites]

I really didn't expect a post on numbers to be so exhausting.

posted by KGMoney at 2:30 AM on April 25, 2017 [1 favorite]

posted by KGMoney at 2:30 AM on April 25, 2017 [1 favorite]

Years ago I got to see Ron Graham give a talk, it was something like Problems that could never be handled computationally. It seemed very clear, he was that exceptional speaker that put these incredibly abstract ideas into a seemingly obvious focus. But like intense pain or the best sex you've ever had, I knew it happened but would never be able to reproduce the feeling, or like actually do the math.

posted by sammyo at 3:22 AM on April 25, 2017 [9 favorites]

posted by sammyo at 3:22 AM on April 25, 2017 [9 favorites]

I enjoyed that right up until my head started to hurt.

I think the INSANITY part came later than I would have put it.

posted by MtDewd at 4:28 AM on April 25, 2017

I think the INSANITY part came later than I would have put it.

posted by MtDewd at 4:28 AM on April 25, 2017

So... lots?

posted by Halloween Jack at 4:28 AM on April 25, 2017

posted by Halloween Jack at 4:28 AM on April 25, 2017

This is a great read and I learned something new. I had never thought of notation past exponents and had also never heard of Graham's number, so yay for learning!

posted by Literaryhero at 4:54 AM on April 25, 2017 [1 favorite]

posted by Literaryhero at 4:54 AM on April 25, 2017 [1 favorite]

Ok I now love this site. Math has never been a strong suit of mine, but I love how this is presented and explained and I stayed with it until about INSANITY, and then I had to reread sections to stay caught up. But, this is a really cool and "simple" way of explaining a math concept. A+ would read again

posted by Suffocating Kitty at 4:57 AM on April 25, 2017 [2 favorites]

posted by Suffocating Kitty at 4:57 AM on April 25, 2017 [2 favorites]

A favorite book I look at once in a while is George Gamow's "One Two Three... Infinity" (with illustrations by the author). Here is his limerick on the inside:

There was a young fellow from Trinity

Who took the square root of Infinity

But the number of digits

Gave him the fidgets

He dropped Math and took up Divinity.

posted by lungtaworld at 5:22 AM on April 25, 2017 [8 favorites]

There was a young fellow from Trinity

Who took the square root of Infinity

But the number of digits

Gave him the fidgets

He dropped Math and took up Divinity.

posted by lungtaworld at 5:22 AM on April 25, 2017 [8 favorites]

I'm not sure if there exists a spigot algorithm that spews out the digits of Graham's number using a constant small amount of RAM.

Someone probably can do it in two lines of Haskell.

Or not.

posted by runcifex at 5:26 AM on April 25, 2017 [1 favorite]

Someone probably can do it in two lines of Haskell.

Or not.

posted by runcifex at 5:26 AM on April 25, 2017 [1 favorite]

Sooo.... think about ggg...(g65 times)...g65

Call that h1

posted by Reverend John at 5:46 AM on April 25, 2017 [1 favorite]

Call that h1

posted by Reverend John at 5:46 AM on April 25, 2017 [1 favorite]

I wanted to read the post before this, from 1 to 1,000,000, but good god. Is he always that misanthropic and misogynistic and generally unlikable? I've only made it to 100 and it's so offputting. Not to mention tired and boring, let's be real. The whole "Let me learn you some fucking shit, dumbass" is overplayed. I felt like I was being mansplained to the whole time.

Which really sucks because I LOVE large numbers.

posted by FirstMateKate at 5:48 AM on April 25, 2017 [6 favorites]

Which really sucks because I LOVE large numbers.

posted by FirstMateKate at 5:48 AM on April 25, 2017 [6 favorites]

Graham's number really is an amazing small number, isn't it? - one of my favourite small numbers, I think. Just incredible to think how much mathematical creativity went in to defining such a tiny little number! "L'il pipsqueak," I call it! No but it's such a tiny little thing, so cute.

OK then - vote #ℵ

posted by the quidnunc kid at 5:56 AM on April 25, 2017 [20 favorites]

OK then - vote #ℵ

_{g64}quidnunc kid!posted by the quidnunc kid at 5:56 AM on April 25, 2017 [20 favorites]

I love this kind of stuff. Factorials scratch the same itch, albeit on a much smaller scale. You can also find ways to drop them into conversation, in a way you probably can't drop Graham's number. For example: shuffling a deck of cards. If you do a really thorough job shuffling, there are 52 * 51 * 50 ... * 2 * 1, or 52! possible arrangements of cards. That's a monstrously large number. It's absurdly unlikely that a deck of cards has ever been arranged in exactly that configuration before, ever, anywhere in the history of the world. How unlikely? Well, factorials grow absurdly fast. 52! is in the neighborhood of 10

Pack a deck of cards and a piece of paper. Every billion years, shuffle and deal yourself a 5-card poker hand. Each time you get a royal flush, buy yourself a PowerBall ticket. If that ticket wins the jackpot, place one grain of sand into the Grand Canyon. When you’ve filled up the canyon with sand, remove one ounce of rock from Mt. Everest. Now empty the canyon and start all over again. When Mt. Everest has been completely removed, make a mark on your piece of paper.

When you get to 256 marks, you've reached 52! seconds.

posted by Mayor West at 5:58 AM on April 25, 2017 [39 favorites]

^{68}, or between "number of grains of sand it would take to fill a hollow planet Earth" and "number of atoms in the observable universe" in this article's scale. That's still not a set of concepts that are really easy to grasp, so let's put that in terms of time and probability. How long is 52! seconds?Pack a deck of cards and a piece of paper. Every billion years, shuffle and deal yourself a 5-card poker hand. Each time you get a royal flush, buy yourself a PowerBall ticket. If that ticket wins the jackpot, place one grain of sand into the Grand Canyon. When you’ve filled up the canyon with sand, remove one ounce of rock from Mt. Everest. Now empty the canyon and start all over again. When Mt. Everest has been completely removed, make a mark on your piece of paper.

When you get to 256 marks, you've reached 52! seconds.

posted by Mayor West at 5:58 AM on April 25, 2017 [39 favorites]

I

posted by lazycomputerkids at 6:05 AM on April 25, 2017 [1 favorite]

*loved*this, and can fight the feeling of being overwhelmed by spotting a typo--*The psycho festival ends when that final feeding frenzy produces***it’s**final number.posted by lazycomputerkids at 6:05 AM on April 25, 2017 [1 favorite]

This head-stretching wankery is fun, but I think Mayor West is on the right track. In real life, the very very inconceivably large numbers are only approached by combinatorics.

posted by whuppy at 6:13 AM on April 25, 2017

posted by whuppy at 6:13 AM on April 25, 2017

That's funny, I always thought Graham's number was

posted by BentFranklin at 6:13 AM on April 25, 2017 [1 favorite]

*Our House*.posted by BentFranklin at 6:13 AM on April 25, 2017 [1 favorite]

Graham's Number is just the start of thinking about big numbers. g

But I think Rayo's number is probably cheating.

Also if you like this there is Googolology Wiki.

posted by graymouser at 6:16 AM on April 25, 2017 [2 favorites]

_{64}<< TREE(3) << SSCG(3).But I think Rayo's number is probably cheating.

Also if you like this there is Googolology Wiki.

posted by graymouser at 6:16 AM on April 25, 2017 [2 favorites]

FirstMateKate, yep, that first part wasn't really good.

Mayor West and whuppy, so true. I've been fascinated by combinatorial arguments, and my obsession is not producing large numbers, but showing how seemingly chaotic terms cancel and sum to a truly small number or zero. There's a very interesting paper on the "DIE" technique of constructive proof. DIE = "Description, Involution, Exception," where the involution is an operation that kills terms pair by pair.

posted by runcifex at 6:28 AM on April 25, 2017 [1 favorite]

Mayor West and whuppy, so true. I've been fascinated by combinatorial arguments, and my obsession is not producing large numbers, but showing how seemingly chaotic terms cancel and sum to a truly small number or zero. There's a very interesting paper on the "DIE" technique of constructive proof. DIE = "Description, Involution, Exception," where the involution is an operation that kills terms pair by pair.

posted by runcifex at 6:28 AM on April 25, 2017 [1 favorite]

size matters.

posted by Obscure Reference at 6:30 AM on April 25, 2017

posted by Obscure Reference at 6:30 AM on April 25, 2017

How long is 52! seconds?

Pack a deck of cards and a piece of paper. Every billion years,

When you get to 256 marks, you've reached 52! seconds.

posted by FirstMateKate at 6:32 AM on April 25, 2017 [16 favorites]

*too damn long, i'll tell you what*Pack a deck of cards and a piece of paper. Every billion years,

**shuffle and deal yourself a 5-card poker hand. Each time you get a royal flush, buy yourself a PowerBall ticket. If that ticket wins the jackpot,***ok, thats it. we've already reached beyond what I can comprehend***place one grain of sand into the Grand Canyon.***If my ticket wins the jackpot, I'm not going to play this game anymore.***When you’ve filled up the canyon with sand,***But you didn't tell me to pack any sand, where am I getting this sand from?***remove one ounce of rock from Mt. Everest.***too big, does not compute***Now empty the canyon and start all over again.***Liquid ounce? weight ounce? I feel like this is the wrong unit of measurment for a mountain.***When Mt. Everest has been completely removed, make a mark on your piece of paper.***This is*__the__worst game of cards..When you get to 256 marks, you've reached 52! seconds.

*but at what cost?*posted by FirstMateKate at 6:32 AM on April 25, 2017 [16 favorites]

Mayor West, well it may be interesting to see 52! is already over 2

posted by runcifex at 6:37 AM on April 25, 2017

^{225}, and 225 bits are already strong enough as an encryption key. Although generating a random key by deck shuffling is neither fast nor very safe..posted by runcifex at 6:37 AM on April 25, 2017

*Years ago I got to see Ron Graham give a talk, it was something like Problems that could never be handled computationally.*

My undergrad Theory of Computation professor (a class which involves studying, in great detail, whether given problems are or are not theoretically computable) used to go off on tangents constantly on what he called "incomputable-computable problems", which are problems that are theoretically computable but the process of computing them would be so large that it couldn't actually be done even if you had a computer that consisted of every atom in the universe.

It was a weird class.

posted by Itaxpica at 6:45 AM on April 25, 2017 [4 favorites]

With a degree in astrophysics, big numbers don't terribly trouble me. Huge powers of 10? Been there, done that. Billions of light-years? Really huge, mind bogglingly huge, but manageable.

When we get to tetration, my brain checks right out. I get the heebie jeebies and start looking for the nearest galactic supercluster to regain my sense of perspective.

posted by tclark at 6:58 AM on April 25, 2017 [12 favorites]

When we get to tetration, my brain checks right out. I get the heebie jeebies and start looking for the nearest galactic supercluster to regain my sense of perspective.

posted by tclark at 6:58 AM on April 25, 2017 [12 favorites]

Numbers go up to 65535, then they start again.

posted by scruss at 7:05 AM on April 25, 2017 [21 favorites]

posted by scruss at 7:05 AM on April 25, 2017 [21 favorites]

There are mathematicians who are questioning the existence of some large numbers, including Graham's number.

posted by Death and Gravity at 7:06 AM on April 25, 2017 [2 favorites]

posted by Death and Gravity at 7:06 AM on April 25, 2017 [2 favorites]

Graham's number is an upper limit.

The actual solution to the problem it constrains might be a lot smaller.

There's a lot of evidence that it's six.

Really.

posted by Combat Wombat at 7:20 AM on April 25, 2017 [19 favorites]

The actual solution to the problem it constrains might be a lot smaller.

There's a lot of evidence that it's six.

Really.

posted by Combat Wombat at 7:20 AM on April 25, 2017 [19 favorites]

*There are mathematicians who are questioning the existence of some large numbers*

I hear their motto is "TB;DC" (too big; didn't count).

posted by the quidnunc kid at 7:50 AM on April 25, 2017 [3 favorites]

**"No one likes pop-ups"**and yet, here you are

posted by Greg Nog at 7:57 AM on April 25, 2017 [4 favorites]

I love that this is the upper bound for this problem is so big that even reading about the notation needed to describe it is enough to provoke existential despair.

For one thing, that is satisfactorily Lovecraftian.

For another, what is the practical use of an upper bound that high? "It is no bigger than... well nothing you can even imagine or wrap your brain around, so if you get that far, stop and turn around. You've passed it."

posted by He Is Only The Imposter at 8:03 AM on April 25, 2017 [5 favorites]

Some people like to play with pain, some people like to put on furry costumes, and now I just found a group of people that like to meta|meta|meta... things.

I not into any of those things, but then I like making gears... so to each his own.

posted by MikeWarot at 8:05 AM on April 25, 2017 [1 favorite]

I not into any of those things, but then I like making gears... so to each his own.

posted by MikeWarot at 8:05 AM on April 25, 2017 [1 favorite]

*There are mathematicians who are questioning the existence of some large numbers, including Graham's number.*

Every sentence in that article reads like it came from the guide in

*The Hitchhiker's Guide to the Galaxy*. Especially this:

Serious work on ultrafinitism has been led, since 1959, by Alexander Esenin-Volpin, who in 1961 sketched a program for proving the consistency of ZFC in ultrafinite mathematics. Other mathematicians who have worked in the topic include Doron Zeilberger, Edward Nelson, and Rohit Jivanlal Parikh. The philosophy is also sometimes associated with the beliefs of Ludwig Wittgenstein, Robin Gandy, and J. Hjelmslev.posted by PlusDistance at 8:11 AM on April 25, 2017 [7 favorites]

Bah, there are more real numbers than that between 0 and 1.

my 5^100 year-old could do that!

posted by thelonius at 8:16 AM on April 25, 2017 [4 favorites]

my 5^100 year-old could do that!

posted by thelonius at 8:16 AM on April 25, 2017 [4 favorites]

You guys are so wrong. One is the largest number.

posted by borkencode at 8:30 AM on April 25, 2017 [2 favorites]

posted by borkencode at 8:30 AM on April 25, 2017 [2 favorites]

Ah, so this is a convenient way of representing the number of evens, in the last few years, that couldn't have been.

posted by bonje at 8:55 AM on April 25, 2017 [8 favorites]

posted by bonje at 8:55 AM on April 25, 2017 [8 favorites]

I use math every day. I advocate for mathematical skills constantly. I've TAed math classes in the past. Compared to the average person, I'm a pretty mathy guy. But. . . this definitely falls into the chasm that separates those of us who merely use math from people who actually relish math. That people are excited by this is both harmless and joyful, and that's great. But, my inner middle-school asshole can't help but retort, "oh yeah, well, 'Graham's number plus one. Woah!'" That one number happens to bound the solution to a completely artificial problem and the other doesn't seems like a pretty arbitrary reason to prefer one number over the other. And my number is bigger, so nyah nyah!

I also very much do want to live forever. I'm not convinced a deeper understanding of forever would change that.

posted by eotvos at 9:50 AM on April 25, 2017

I also very much do want to live forever. I'm not convinced a deeper understanding of forever would change that.

posted by eotvos at 9:50 AM on April 25, 2017

I think if you lived forever, you'd end up being a creature of the (relative) moment, because memories themselves become problematic. Ideally, you would have a decent short-term memory, of, say, a few dozens of years, and an excellent "muscle" or "instinctual" memory where as you acquired new skills or knowledge they replaced the inferior version that proceeded them.

Or you could just wander around and insult every sentient being, in alphabetical order.

posted by maxwelton at 10:05 AM on April 25, 2017 [6 favorites]

Or you could just wander around and insult every sentient being, in alphabetical order.

posted by maxwelton at 10:05 AM on April 25, 2017 [6 favorites]

Weird, I got to the tetration part and when i grasped the 3^3^3^3 part I got really emotional and sort of choked up for a second.

posted by Stonestock Relentless at 10:07 AM on April 25, 2017 [3 favorites]

posted by Stonestock Relentless at 10:07 AM on April 25, 2017 [3 favorites]

Not unrelated to this is the extremely nerdy BigNum Bakeoff.

Summary for those who hate links: You write a program that generates a number. Biggest number wins. You are using a completely theoretical computer and a modified version of C that has an "int" type that can hold arbitrarily large numbers.

Some people used variations of 9^9^9^9^... Some people used variations of the power tower described above. These produce numbers that are, respectively, "tiny" and "small". To get "big" and "really big" you need a completely different bag of tricks.

One day I will understand everything in that article. Today is not that day.

posted by It's Never Lurgi at 10:10 AM on April 25, 2017 [3 favorites]

Summary for those who hate links: You write a program that generates a number. Biggest number wins. You are using a completely theoretical computer and a modified version of C that has an "int" type that can hold arbitrarily large numbers.

Some people used variations of 9^9^9^9^... Some people used variations of the power tower described above. These produce numbers that are, respectively, "tiny" and "small". To get "big" and "really big" you need a completely different bag of tricks.

One day I will understand everything in that article. Today is not that day.

posted by It's Never Lurgi at 10:10 AM on April 25, 2017 [3 favorites]

*Ok I now love this site.*

His articles on Why Procrastinators Procrastinate and How to Beat Procrastination are really great and possibly not unrelated to the existence of the article about Graham's number or comment you are currently reading.

posted by straight at 10:20 AM on April 25, 2017 [1 favorite]

For "really big", I suspect you need the Busy Beaver as in theory (but not so much in practice because of the size of the thing), Graham's number is computable by a Turing Machine of some size N. BusyBeaver(N) can't be computed by such a TM of size N, but is still finite and larger than Graham's number.

posted by Death and Gravity at 10:20 AM on April 25, 2017

posted by Death and Gravity at 10:20 AM on April 25, 2017

For another excellent take on "the biggest number" see this article.

posted by grog at 10:38 AM on April 25, 2017 [1 favorite]

posted by grog at 10:38 AM on April 25, 2017 [1 favorite]

If you get bored of large finite numbers, there's a whole other world of countable infinities to blow your mind: John Carlos Baez on Large Countable Ordinals Part 1 / Part 2 / Part 3

posted by panic at 10:55 AM on April 25, 2017

posted by panic at 10:55 AM on April 25, 2017

Thanks, but I'll just content myself with hrair.

posted by Winnie the Proust at 12:26 PM on April 25, 2017 [1 favorite]

posted by Winnie the Proust at 12:26 PM on April 25, 2017 [1 favorite]

Death and Gravity beat me to commenting on the real winner of the big numbers contest, the busy beaver function BB(N). Very roughly, that's the maximum number of calculation steps that can be taken by any computing device with N internal states that doesn't just run forever.

Think about the description in the article of going from g1 to Graham's number. BB(12) is bigger than g1 (mentioned in the article), and BB(23) is bigger than Graham's number. But much more interestingly, it's recently been proven that BB(7918)

posted by madmethods at 12:49 PM on April 25, 2017 [5 favorites]

Think about the description in the article of going from g1 to Graham's number. BB(12) is bigger than g1 (mentioned in the article), and BB(23) is bigger than Graham's number. But much more interestingly, it's recently been proven that BB(7918)

*can't be calculated using our rules of mathematics at all*. That 7918 number is also an upper bound; most people familiar with this stuff seem to think that the actual value where things break down is much lower. So the function grows so fast that at some specific number that is totally within human understanding (likely much less than 7918) it bangs into a deep limit of mathematics and computation.posted by madmethods at 12:49 PM on April 25, 2017 [5 favorites]

A deep limit of mathematics and computation? As if I couldn't be messed with more.

posted by Stonestock Relentless at 1:45 PM on April 25, 2017

posted by Stonestock Relentless at 1:45 PM on April 25, 2017

*But much more interestingly, it's recently been proven that BB(7918) can't be calculated using our rules of mathematics at all. That 7918 number is also an upper bound; most people familiar with this stuff seem to think that the actual value where things break down is much lower.*

The really astonishing thing there is that BB(

*5*) is still unknown. It's at least 4098, but there are still about 40 5-state machines for which are suspected, but not proven, to never halt. There are Turing machines with just

*5 states and 2 symbols*for which it's still unknown whether they halt.

posted by DevilsAdvocate at 2:11 PM on April 25, 2017 [1 favorite]

It's interesting you can do all this and more starting from the numbers we know, and still not break out of the finite, which is why (some) mathematicians describe the cardinality of the natural numbers (ℵ

posted by jamjam at 3:29 PM on April 25, 2017 [3 favorites]

_{0}), as an inaccessible cardinal:In set theory, an uncountable cardinal is inaccessible if it cannot be obtained from smaller cardinals by the usual operations of cardinal arithmetic. ...It strikes me as a little strange that on the other side of ℵ

The term "inaccessible cardinal" is ambiguous. Until about 1950 it meant "weakly inaccessible cardinal", but since then it usually means "strongly inaccessible cardinal". An uncountable cardinal is weakly inaccessible if it is a regular weak limit cardinal. It is strongly inaccessible, or just inaccessible, if it is a regular strong limit cardinal (this is equivalent to the definition given above). Some authors do not require weakly and strongly inaccessible cardinals to be uncountable (in which case ℵ_{0}is strongly inaccessible). Weakly inaccessible cardinals were introduced by Hausdorff (1908), and strongly inaccessible ones by Sierpiński & Tarski (1930) and Zermelo (1930). ...

_{0}, you can generate new transfinite cardinals in profusion with various operations on cardinals, but ZFC (Zermelo-Frankel set theory + the Axiom of Choice) cannot prove or disprove the existence of another inaccessible cardinal, but if you assume the existence of an inaccessible cardinal, you get the really huge Grothendiek sets, which, as far as I've heard, are still a necessary element in all extant proofs of Fermat's Last Theorem. It's almost as if you could have a hierarchy of inaccessible cardinals starting with ℵ_{0}, and beneath each of them you would have a distinct realm of cardinal numbers all but the first of which would be transfinite, but which might have parallels which would make it possible to have a relatively easy proof in one translate into a proof that seemed much harder in another.posted by jamjam at 3:29 PM on April 25, 2017 [3 favorites]

The original article made my brain hurt but

posted by Johnny Wallflower at 5:07 PM on April 25, 2017

**jamjam**'s comment made it melt and run out of my ear holes. Thanks a lot.posted by Johnny Wallflower at 5:07 PM on April 25, 2017

This really brings it home how, like the reals, the set of natural numbers is 0% accessible via finite expressions. With a finite expression we can write down arbitrarily large numbers, or exceed any given number, but once you get into "more digits than you could ever represent in the observable universe" territory, there are necessarily vast gulfs of unrepresentable numbers.

(Probably the wrong site for this question, but is there any reason why ordering is even finitely decidable for finite expressions? I.e., are there scary hyper-iterated operations F and G for which F(x,G(x)) < G(x,F(x)) can never be determined in the age of the universe?)

posted by mubba at 6:54 PM on April 25, 2017

(Probably the wrong site for this question, but is there any reason why ordering is even finitely decidable for finite expressions? I.e., are there scary hyper-iterated operations F and G for which F(x,G(x)) < G(x,F(x)) can never be determined in the age of the universe?)

posted by mubba at 6:54 PM on April 25, 2017

*BB(12) is bigger than g1 (mentioned in the article), and BB(23) is bigger than Graham's number.*

Far more mind-blowing to me than either the article about Graham's number or the articles about Busy Beavers is the idea that it's possible to determine whether BB(23) is bigger than Graham's number.

It's one thing to describe an arbitrarily-large number with paragraph after paragraph of "and then take the whole thing and do THIS, but then do it THAT many times," but to actually

*use*those numbers mathematically...

posted by straight at 8:05 AM on April 26, 2017

> Remember ∞ is not a number.

Yes, but ω is! And so is ω2! And while ω != ω2, 2ω = ω!

(The commutative property is a happy accident for finite numbers, and completely falls apart for transfinite numbers because multiplication in that domain behaves like vectors; eg: x T y (which makes a kind of sense when you define integers as ordered sets of the empty set.) )

posted by Xyanthilous P. Harrierstick at 1:14 PM on April 26, 2017

Yes, but ω is! And so is ω2! And while ω != ω2, 2ω = ω!

(The commutative property is a happy accident for finite numbers, and completely falls apart for transfinite numbers because multiplication in that domain behaves like vectors; eg: x T y (which makes a kind of sense when you define integers as ordered sets of the empty set.) )

posted by Xyanthilous P. Harrierstick at 1:14 PM on April 26, 2017

I believe that there are only three classes of integers. There are small integers, for countable things. The number of books is a small integer; so is the number of hairs on someone's head. There are medium integers for things that are too numerous to count, but which can be expressed as an order of magnitude: the number of atoms in our planet, for instance. And then there are things that are even too large to express as an order of magnitude, but which we can use asan item of comparison: the stopping time of some Busy Beavers may be like the number of quantum states in the universe.

I can't even imagine what a larger class might be like.

posted by Joe in Australia at 4:38 PM on April 26, 2017

I can't even imagine what a larger class might be like.

posted by Joe in Australia at 4:38 PM on April 26, 2017

*Far more mind-blowing to me than either the article about Graham's number or the articles about Busy Beavers is the idea that it's possible to determine whether BB(23) is bigger than Graham's number.*

Probably not as mind-blowing as it sounds. If you can encode the computation of Graham's number into a 23 state Turing Machine then you have proven that BB(23) is at least as big as Graham's number.

posted by It's Never Lurgi at 4:55 PM on April 27, 2017

Really there are two kinds of integers, cardinals which tell you how many things there are, and ordinals which provide order to those things. And they're not at all the same unless they are finite. There are huge ordinals that are still only countable (that is less than or equal to ℵ

posted by Death and Gravity at 6:10 PM on April 27, 2017

_{0}- the cardinal of the integers - see the Joan Baez link above).posted by Death and Gravity at 6:10 PM on April 27, 2017

*...And then there are things that are even too large to express as an order of magnitude, but which we can use asan item of comparison: the stopping time of some Busy Beavers may be like the number of quantum states in the universe.*

I can't even imagine what a larger class might be like.

I can't even imagine what a larger class might be like.

I'm afraid that larger class is exactly what we're talking about here -- Grahahm's number and these other numbers are vastly, incomprehensibly larger than the number of quantum states in the universe.

posted by madmethods at 8:09 AM on April 28, 2017

*If you can encode the computation of Graham's number into a 23 state Turing Machine then you have proven that BB(23) is at least as big as Graham's number.*

Someone has encoded the computation of Graham's number into a Turing Machine? Or did they just somehow demonstrate that it's

*possible*to encode Graham's number into a 23 state Turning Machine?

posted by straight at 2:48 PM on April 28, 2017

graymouser: "

I love TREE(3), and love it even more now that the OP article has shed some light on how tetration works.

As Wiki describes it:

*Graham's Number is just the start of thinking about big numbers. g64 << TREE(3)*"I love TREE(3), and love it even more now that the OP article has shed some light on how tetration works.

As Wiki describes it:

The TREE sequence begins TREE(1) = 1, TREE(2) = 3, then suddenly TREE(3) explodes to a value so enormously large that many other "large" combinatorial constants, such as Friedman's n(4),[*] are extremely small by comparison. A lower bound for n(4), and hence an extremely weak lower bound for TREE(3), is A(A(...A(1)...)), where the number of As is A(187196), and A() is a version of Ackermann's function: A(x) = 2 [x+1] x in hyperoperation. Graham's number, for example, is approximately A64(4) which is much smaller than the lower bound AA(187196)(1).posted by Rhaomi at 11:35 PM on April 28, 2017

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I just got my daughter "How Much Is a Million" which is like the first few paragraphs of this blog post in children's book form.

posted by potrzebie at 12:15 AM on April 25, 2017 [2 favorites]