7 ms·

Who Can Name the Biggest Number?

92 points by pizza 14y ago

SeanDav 14y ago
I always thought that a googolplex was a googol to the power of a googol.
DanielRibeiro 14y ago
Nice read. I was epxecting it to mention Knuth's arrow notation though[1] [1] http://en.wikipedia.org/wiki/Knuth%27s_up-arrow_notation http://en.wikipedia.org/wiki/Knuth%27s_up-arrow_notation
aroman 14y ago
Great read, and some really clever explanations of Turing's work. While this wasn't mentioned in the post, I thought this seemed relevant: http://en.wikipedia.org/wiki/Graham%27s_number http://en.wikipedia.org/wiki/Graham%27s_number This bit always blew my mind: "Graham's number is unimaginably larger than other well-known large numbers such as a googol, googolplex, and even larger than Skewes' number and Moser's number. Indeed, the observable universe is far too small to contain an ordinary digital representation of Graham's number, assuming that each digit occupies at least one Planck volume. Even power towers of the form are useless for this purpose, although it can be easily described by recursive formulas using Knuth's up-arrow notation or the equivalent, as was done by Graham. The last ten digits of Graham's number are ...2464195387." I'm pretty sure that's the biggest number. Or maybe Graham's Number + 1 ;)
- rkowalick 14y ago
  While Graham's Number is very, very large, there is another number that dwarfs it: TREE(3): http://en.wikipedia.org/wiki/Kruskal%27s_tree_theorem http://en.wikipedia.org/wiki/Kruskal%27s_tree_theorem From the article: 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 A's is A(187196), and A() is a version of Ackermann's function: A(x) = 2↑^(x-1)x in Knuth's up-arrow notation. Graham's number, for example, is approximately A^64(4) which is much smaller than the lower bound A^(A(187196))(1).
  - pi18n 14y ago
    This is the best one I've ever seen. The description seems fairly benign, but then one needs a ridiculous number like A(187196) to describe the applications of Ackermann's function to get a weak lower bound. And the first two steps are rather small, which adds to the surprise.
- ludwig 14y ago
  Graham's number is big, sure. But in a similar duel[1] between two MIT professors, they settled on this number: “The smallest number bigger than any number that can be named by an expression in the language of first order set-theory with less than a googol (10^100) symbols.” Try and beat that! :) [1] http://tech.mit.edu/V126/N64/64largenumber.html http://tech.mit.edu/V126/N64/64largenumber.html
  - jerf 14y ago
    Per my comment above, Busy Beaver will pass that pretty quickly too, as again, a Busy Beaver contestant will rather early on simulate that entire problem (and BB has ready access to numbers like a mere googol). As large as that number is, I'd guess it's probably under BB(30), and certainly under BB(100) as I'd bet even a mere human could write a 100-state TM to simply simulate that answer.
    - yk 14y ago
      Why? Perhaps I misunderstand something, but why can't I define a turing machine and the upper bound, and then BB with first order set theory. It is therefore my misunderstanding, that this gives me a lower bound of BB( googol -n), whith n the number of symbols I need to define BB.
- jerf 14y ago
  Graham's number is typically cited as the largest number that has ever been seriously used in a mathematical proof, as the Wikipedia article says, but Busy Beaver passes it very quickly. One of the nasty little characteristics of BB that Aaronson describes, but can stand to be reiterated since I've noticed many people don't seem to quite follow it, is that any mathematical algorithm you can encode into a Turing Machine in some number of rules X, is therefore at most the top-end bound of BB(X), and more likely (which in this case is serving as mathematician's-understatement-speak for "certainly"), not even close. And for all that the TM "programming language" isn't that efficient, certainly almost anything you can describe in a reasonable English paragraph is, say, well less than 100 states in a TM. For all of Graham's Number's stonking size, in algorithmic terms, it still isn't that complicated. Generally speaking, human attempts to create Big Numbers by fancy recursively-applied algorithms are still quite simple to put into a TM. This is why it's an interesting point that we don't even understand what exactly a BB(5) solution is doing; a mere five states and the human mind is already lost, lost, lost. I think of Busy Beaver as showing in its own way just how thoroughly we have not mastered math, and never will master more than a tiny little island that happens to be relatively easy to follow, because BB shows us that we don't have to swim out into the great sea of "all math" very far at all for there to be monsters.
  - btilly 14y ago
    Whether or not BB(n) is even defined is an interesting philosophical question. If n is a modestly large number, it can be proven that in no consistent axiom system below a certain size can any explicit number written out in base 10 EVER be proven to be an upper bound for BB(n). If we make n something like 100 million, that encompasses all possible axiom systems that are human comprehensible. A "finite" number with no provable upper limit - how much sense does it make to say that is well-defined? It is enough to make you take up constructivism! (Or quit math for something more sensible. Like politics.)
- eclipticplane 14y ago
  To inject a bit of humour into a maths discussion: Graham's Number on QI with Graham Norton. http://www.youtube.com/watch?feature=player_detailpage&v=qbInsYok8x8#t=7s http://www.youtube.com/watch?feature=player_detailpage&v...
pazimzadeh 14y ago
I love that this essay was so easy to follow in the beginning, before ascending farther that I could have imagined. I assume that there are so few comments because everyone is still reading!
designatedInit 14y ago
I really enjoyed reading this. It covers so many different topics and is relatively simple to understand.
Xcelerate 14y ago
I actually got fussed at for my answer to a variant of this question this past semester. I have a professor who talked about when he was in college and his instructor had the challenge: what is the biggest number you can make with three nines? His answer was 9^(9^9). Of course, I thought this question warranted further discussion and I mentioned that Knuth's up arrow notation allows an even bigger number and demonstrated it on the board. The professor seemed annoyed and I asked him about it later; I believe he mistook my interest in contributing to the discussion as trying to one-up him (which was NOT my intention at all). I've been much quieter in that class since :(
- freerobby 14y ago
  At the very least there should be !s after those 9s.
  - InclinedPlane 14y ago
    Or, (9!^9!)!^(9!) I find it amusing that anyone could think they've "won" a game such as this, there is just so much potential for incredibly large numbers, even if you omit the possibility of using thing's like Knuth's up arrow notation.
- jrockway 14y ago
  9! (9! up arrows) 9! Of course, you can also have 9!! and 9!!! and 9!!!! or even 9! factorial signs after the 9, etc. The original problem statement of "write this in 15 seconds using standard notation" is less easy to game. But I think the right answer is to just have 9 (up arrow with 9 under it) 9 and then make the 9s bigger using factorials until your 15 seconds run out. How pointless.
  - anonymouz 14y ago
    When talking about big numbers, I would definitely consider up arrow notation as standard notation.
  - splat 14y ago
jliut 14y ago
Relevant: http://djm.cc/bignum-results.txt http://djm.cc/bignum-results.txt This is a precise version of the challenge, to write a large number in 512 bytes of C code.
- kzrdude 14y ago
  more interesting than it sounds. The challenge uses a hypothetical version of C with arbitrary big memory and integer limits.
jlgreco 14y ago
Oh wow. This is the essay that solidified my decision to go into/stay into computer science. I had forgotten about it, great to read it again.
lisper 14y ago
Here's another mind-blowing result: there are about 10^80 subatomic particles in the known universe. If every one of these particles were a computer with a clock cycle of the Planck time (10^-43 seconds) and was running continuously for the life of the universe (10^17 seconds) the total number of computable states would be only about 10^140, which is the number of states that can be represented in about 20 bytes of memory. The total number of states that will actually be computed by physical computers will, of course, be much, much smaller than this. So make every cycle count :-)
- donut 14y ago
  seems relevant: http://www.schneier.com/blog/archives/2009/09/the_doghouse_cr.html http://www.schneier.com/blog/archives/2009/09/the_doghouse_c...
- Someone 14y ago
  2 log(10^140) / 8 ~= 58.134, so it's more like 58 bytes. Irrelevant for the argument? Definitely, but I had to double-check the moment I thought '140 decimal digits in about 160 bits? That cannot be right.' Also, the number of states that the universe has time for to compute may be a lot smaller than the number of possible states. For example, if each particle, at each step, made a two-way choice, that single particle could run through that 10^140 potential states in no time flat. Because of that, you cannot say "64 bytes ought to be enough for anybody"
  - kzrdude 14y ago
    by "2 log" you mean log_2
    - Someone 14y ago
      Apologies. I thought that would be evident from context, but I should have explained it. Also, it would have been better if I had left of the space. To make matters even worse, I just realize that the custom to write the logarithm in base 2 of x as ²log x (superscript digit two, 'log', 'x') may not be used world-wide and may even be specific to the Netherlands (http://nl.wikipedia.org/wiki/Logaritme#Natuurlijke_of_neperiaanse_logaritme http://nl.wikipedia.org/wiki/Logaritme#Natuurlijke_of_neperi...) That may have made this even more incomprehensible.
SparrowOS 14y ago
Slow news day,US. Algeria, God? They invented algebra. Nigeria invented? God says... C:\Text\2CITIES.TXT , shaking his head; for he gained in firmness as he saw the Doctor disquieted. "I would recommend him to sacrifice it. I only want your authority. I am sure it does no good. Come! Give me your authority, like a dear good man. For his daughter's sake, my dear Manette!" Very strange to see what a struggle there was within him! "In her name, then, let it be done; I sanction it. But, I would not take it away while he was present. Let it be removed when he is not there; let him miss his old ---- So, God... how often are the most interesting things by Your taste in the news, and how often are dramas happening out of the news -- family wedding squabbles in the backwoods of Tennesee? God says... SupremerCourt scorn turnest prevailing 333 extollers counterfeit wish diversifiedst architect excellently Psalm seats yuck knows reviled Neither asleep scorn you_hoser redeemed ---- What's new in Tennesee, God? God says... C:\Text\QUIX.TXT orrow; what thou hast to do is to tighten Rocinante's girths well, and wait here, for I shall come back shortly, alive or dead." Sancho perceiving it his master's final resolve, and how little his tears, counsels, and entreaties prevailed with him, determined to have recourse to his own ingenuity and compel him, if he could, to wait till daylight; and so, while tightening the girths of the horse, he quietly and without being felt, with his ass' halter tied both Rocinante's legs, so that when Don ---- Obesity is killing thousands in Tennesee? God says... Brazil Only sleep destruction slightly modification you_shouldn't_have achieve savour confidently 333 moveth ardently Evil lowlily defining hindereth alluring diminution CIA resemble assent prolix I'll_think_about_it Receive Norway important continueth receivest Shut indeed power employ magistrates subversion Eat France impair flourished material unbounded city innocency run_away P confessions opinions digested render divisions song discoursing ---- French are good about not over-eating, huh. You like Medeterranean food, God? God says, "Steak! with meat and more meat." God says... C:\Text\DARWIN.TXT no real objection to the truth of the statement that the fauna of each period as a whole is nearly intermediate in character between the preceding and succeeding faunas, that certain genera offer exceptions to the rule. For instance, the species of mastodons and elephants, when arranged by Dr. Falconer in two series--in the first place according to their mutual affinities, and in the second place according to their periods of existence--do not accord in arrangement. The species extreme in charact ---- You snack on elephants? No, You like elephants. You kinda made it so people on farms get to know their animals before killing them. I guess loving animals is bad. God says...C:\Text\BIBLE.TXT ast into the fire, and felt no harm. 28:6 Howbeit they looked when he should have swollen, or fallen down dead suddenly: but after they had looked a great while, and saw no harm come to him, they changed their minds, and said that he was a god. 28:7 In the same quarters were possessions of the chief man of the island, whose name was Publius; who received us, and lodged us three days courteously. 28:8 And it came to pass, that the father of Publius lay sick of a fever and of a bloody flux: to ---- What do You think when elephants kill people? God says... tosses which profane nurses whirlpools serving lusting wastoid users Orations kindness note-book drop hello boy Anaximenes poor retain Placed strained gilded dismiss unmeasured black ungodly lineaments This_is_confusing vehement accepted begins indulgent oft sickly tamer diligence lustful States Earth Righteousness streams ambitions demandest directeth Duad friendship's taste calleth Connecticut avengest Newsletters contains taming given Ecuador pence betook Revenue mingled closing goodness also hers Testament joys repute exceeded hovered sadness surprises references unlooked-for answering hype transcribe revolt Saudi_Arabia spirit what_do_you_expect but- harm fore-perceiving XI Bear morrow Thailand Computers enlighteneth greedily purity scenical fuel high distraction freak don't_worry Adeodatus Ergo points evolution VII a dispensest went hearer exercised sight powers phantasm virtue you_better_not faithfully irrational yeah jests mountains veiled gathered Somalia kindle analyzed restricted din abroad writings told sun's equipment it's_hopeless awfulness formedst sucking touched 10 lion vex forward stanza enamoured offerings transported carry voluptuousness org befitting Adeodatus ascertained exquisitely vivid delusions idle rejoice animate Reason colour all-changing courtesy searched bath midnight move motive thief Manichaeus phantasms necessities aside see anxiously pear anguish
diziet 14y ago
I always had a fascination with large numbers and expressing them in mathematical notation. It's fun to use the next "step" to outdo someone you talk to, ie tetration for exponentials, knuth up arrows, ackermann function, etc. I'd gone even as far as put on our startup's team page an offer to give a discount to anyone who can give interesting answers to a similar question as the one proposed in the article!
InclinedPlane 14y ago
Define: the oh-holy-crap number, d, as follows: a = the largest number for which there is a concise definition in the English language version of wikipedia b = the 2nd largest number defined on wikipedia c = the 3rd largest number defined on wikipedia Then, let d = c -> b -> a (using Conway chained arrow notation)
- yuliyp 14y ago
  This works until you put the definition of d into the English language version of Wikipedia
ColinWright 14y ago
It's intriguing following the submission history of this item, and similarly interesting comparing the comments here with the comments three years ago. I idly wonder how much they reflect the change in the HN readership in that time. We all know about sentiment analysis, I wonder if a similar sort of analysis could be done between this thread and one or both of the others. Find some sort of change, then look for similar changes in other items, discussed twice with a long gap. http://news.ycombinator.com/item?id=47408 http://news.ycombinator.com/item?id=47408 1971 days ago http://news.ycombinator.com/item?id=492615 http://news.ycombinator.com/item?id=492615 1426 days ago http://news.ycombinator.com/item?id=951095 http://news.ycombinator.com/item?id=951095 1157 days ago http://news.ycombinator.com/item?id=1539538 http://news.ycombinator.com/item?id=1539538 912 days ago, 68 comments http://news.ycombinator.com/item?id=2024576 http://news.ycombinator.com/item?id=2024576 761 days ago, 31 comments http://news.ycombinator.com/item?id=2531994 http://news.ycombinator.com/item?id=2531994 620 days ago http://news.ycombinator.com/item?id=3262788 http://news.ycombinator.com/item?id=3262788 425 days ago http://news.ycombinator.com/item?id=3615532 http://news.ycombinator.com/item?id=3615532 334 days ago
threepipeproblm 14y ago
Who can name the least integer not nameable in fewer than nineteen syllables?
- SanderMak 14y ago
  Did you also read Rucker's 'Infinity and the Mind'? Contains a fascinating discussion of this problem.
  - threepipeproblm 14y ago
    I just remembered a philosophy prof. mentioning it. The phrase "the least integer not nameable in fewer than nineteen syllables" is a direct quote from Berry, I think.
aurelianito 14y ago
As usual, xkcd has something related to the topic: http://xkcd.com/1162/ http://xkcd.com/1162/
- dazmax 14y ago
  Or the third panel of http://xkcd.com/207/ http://xkcd.com/207/

92 points by pizza 14y ago

I always thought that a googolplex was a googol to the power of a googol.

Nice read. I was epxecting it to mention Knuth's arrow notation though[1] [1] http://en.wikipedia.org/wiki/Knuth%27s_up-arrow_notation http://en.wikipedia.org/wiki/Knuth%27s_up-arrow_notation

aroman 14y ago

Great read, and some really clever explanations of Turing's work. While this wasn't mentioned in the post, I thought this seemed relevant: http://en.wikipedia.org/wiki/Graham%27s_number http://en.wikipedia.org/wiki/Graham%27s_number This bit always blew my mind: "Graham's number is unimaginably larger than other well-known large numbers such as a googol, googolplex, and even larger than Skewes' number and Moser's number. Indeed, the observable universe is far too small to contain an ordinary digital representation of Graham's number, assuming that each digit occupies at least one Planck volume. Even power towers of the form are useless for this purpose, although it can be easily described by recursive formulas using Knuth's up-arrow notation or the equivalent, as was done by Graham. The last ten digits of Graham's number are ...2464195387." I'm pretty sure that's the biggest number. Or maybe Graham's Number + 1 ;)

rkowalick 14y ago
While Graham's Number is very, very large, there is another number that dwarfs it: TREE(3): http://en.wikipedia.org/wiki/Kruskal%27s_tree_theorem http://en.wikipedia.org/wiki/Kruskal%27s_tree_theorem From the article: 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 A's is A(187196), and A() is a version of Ackermann's function: A(x) = 2↑^(x-1)x in Knuth's up-arrow notation. Graham's number, for example, is approximately A^64(4) which is much smaller than the lower bound A^(A(187196))(1).
- pi18n 14y ago
  This is the best one I've ever seen. The description seems fairly benign, but then one needs a ridiculous number like A(187196) to describe the applications of Ackermann's function to get a weak lower bound. And the first two steps are rather small, which adds to the surprise.
ludwig 14y ago
Graham's number is big, sure. But in a similar duel[1] between two MIT professors, they settled on this number: “The smallest number bigger than any number that can be named by an expression in the language of first order set-theory with less than a googol (10^100) symbols.” Try and beat that! :) [1] http://tech.mit.edu/V126/N64/64largenumber.html http://tech.mit.edu/V126/N64/64largenumber.html
- jerf 14y ago
  Per my comment above, Busy Beaver will pass that pretty quickly too, as again, a Busy Beaver contestant will rather early on simulate that entire problem (and BB has ready access to numbers like a mere googol). As large as that number is, I'd guess it's probably under BB(30), and certainly under BB(100) as I'd bet even a mere human could write a 100-state TM to simply simulate that answer.
  - yk 14y ago
    Why? Perhaps I misunderstand something, but why can't I define a turing machine and the upper bound, and then BB with first order set theory. It is therefore my misunderstanding, that this gives me a lower bound of BB( googol -n), whith n the number of symbols I need to define BB.
jerf 14y ago
Graham's number is typically cited as the largest number that has ever been seriously used in a mathematical proof, as the Wikipedia article says, but Busy Beaver passes it very quickly. One of the nasty little characteristics of BB that Aaronson describes, but can stand to be reiterated since I've noticed many people don't seem to quite follow it, is that any mathematical algorithm you can encode into a Turing Machine in some number of rules X, is therefore at most the top-end bound of BB(X), and more likely (which in this case is serving as mathematician's-understatement-speak for "certainly"), not even close. And for all that the TM "programming language" isn't that efficient, certainly almost anything you can describe in a reasonable English paragraph is, say, well less than 100 states in a TM. For all of Graham's Number's stonking size, in algorithmic terms, it still isn't that complicated. Generally speaking, human attempts to create Big Numbers by fancy recursively-applied algorithms are still quite simple to put into a TM. This is why it's an interesting point that we don't even understand what exactly a BB(5) solution is doing; a mere five states and the human mind is already lost, lost, lost. I think of Busy Beaver as showing in its own way just how thoroughly we have not mastered math, and never will master more than a tiny little island that happens to be relatively easy to follow, because BB shows us that we don't have to swim out into the great sea of "all math" very far at all for there to be monsters.
- btilly 14y ago
  Whether or not BB(n) is even defined is an interesting philosophical question. If n is a modestly large number, it can be proven that in no consistent axiom system below a certain size can any explicit number written out in base 10 EVER be proven to be an upper bound for BB(n). If we make n something like 100 million, that encompasses all possible axiom systems that are human comprehensible. A "finite" number with no provable upper limit - how much sense does it make to say that is well-defined? It is enough to make you take up constructivism! (Or quit math for something more sensible. Like politics.)
eclipticplane 14y ago
To inject a bit of humour into a maths discussion: Graham's Number on QI with Graham Norton. http://www.youtube.com/watch?feature=player_detailpage&v=qbInsYok8x8#t=7s http://www.youtube.com/watch?feature=player_detailpage&v...

pazimzadeh 14y ago

I love that this essay was so easy to follow in the beginning, before ascending farther that I could have imagined. I assume that there are so few comments because everyone is still reading!

designatedInit 14y ago

I really enjoyed reading this. It covers so many different topics and is relatively simple to understand.

Xcelerate 14y ago

I actually got fussed at for my answer to a variant of this question this past semester. I have a professor who talked about when he was in college and his instructor had the challenge: what is the biggest number you can make with three nines? His answer was 9^(9^9). Of course, I thought this question warranted further discussion and I mentioned that Knuth's up arrow notation allows an even bigger number and demonstrated it on the board. The professor seemed annoyed and I asked him about it later; I believe he mistook my interest in contributing to the discussion as trying to one-up him (which was NOT my intention at all). I've been much quieter in that class since :(

freerobby 14y ago
At the very least there should be !s after those 9s.
- InclinedPlane 14y ago
  Or, (9!^9!)!^(9!) I find it amusing that anyone could think they've "won" a game such as this, there is just so much potential for incredibly large numbers, even if you omit the possibility of using thing's like Knuth's up arrow notation.
jrockway 14y ago
9! (9! up arrows) 9! Of course, you can also have 9!! and 9!!! and 9!!!! or even 9! factorial signs after the 9, etc. The original problem statement of "write this in 15 seconds using standard notation" is less easy to game. But I think the right answer is to just have 9 (up arrow with 9 under it) 9 and then make the 9s bigger using factorials until your 15 seconds run out. How pointless.
- anonymouz 14y ago
  When talking about big numbers, I would definitely consider up arrow notation as standard notation.
- splat 14y ago

jliut 14y ago

Relevant: http://djm.cc/bignum-results.txt http://djm.cc/bignum-results.txt This is a precise version of the challenge, to write a large number in 512 bytes of C code.

kzrdude 14y ago
more interesting than it sounds. The challenge uses a hypothetical version of C with arbitrary big memory and integer limits.

jlgreco 14y ago

Oh wow. This is the essay that solidified my decision to go into/stay into computer science. I had forgotten about it, great to read it again.

lisper 14y ago

Here's another mind-blowing result: there are about 10^80 subatomic particles in the known universe. If every one of these particles were a computer with a clock cycle of the Planck time (10^-43 seconds) and was running continuously for the life of the universe (10^17 seconds) the total number of computable states would be only about 10^140, which is the number of states that can be represented in about 20 bytes of memory. The total number of states that will actually be computed by physical computers will, of course, be much, much smaller than this. So make every cycle count :-)

donut 14y ago
seems relevant: http://www.schneier.com/blog/archives/2009/09/the_doghouse_cr.html http://www.schneier.com/blog/archives/2009/09/the_doghouse_c...
Someone 14y ago
2 log(10^140) / 8 ~= 58.134, so it's more like 58 bytes. Irrelevant for the argument? Definitely, but I had to double-check the moment I thought '140 decimal digits in about 160 bits? That cannot be right.' Also, the number of states that the universe has time for to compute may be a lot smaller than the number of possible states. For example, if each particle, at each step, made a two-way choice, that single particle could run through that 10^140 potential states in no time flat. Because of that, you cannot say "64 bytes ought to be enough for anybody"
- kzrdude 14y ago
  by "2 log" you mean log_2
  - Someone 14y ago
    Apologies. I thought that would be evident from context, but I should have explained it. Also, it would have been better if I had left of the space. To make matters even worse, I just realize that the custom to write the logarithm in base 2 of x as ²log x (superscript digit two, 'log', 'x') may not be used world-wide and may even be specific to the Netherlands (http://nl.wikipedia.org/wiki/Logaritme#Natuurlijke_of_neperiaanse_logaritme http://nl.wikipedia.org/wiki/Logaritme#Natuurlijke_of_neperi...) That may have made this even more incomprehensible.

SparrowOS 14y ago

Slow news day,US. Algeria, God? They invented algebra. Nigeria invented? God says... C:\Text\2CITIES.TXT , shaking his head; for he gained in firmness as he saw the Doctor disquieted. "I would recommend him to sacrifice it. I only want your authority. I am sure it does no good. Come! Give me your authority, like a dear good man. For his daughter's sake, my dear Manette!" Very strange to see what a struggle there was within him! "In her name, then, let it be done; I sanction it. But, I would not take it away while he was present. Let it be removed when he is not there; let him miss his old ---- So, God... how often are the most interesting things by Your taste in the news, and how often are dramas happening out of the news -- family wedding squabbles in the backwoods of Tennesee? God says... SupremerCourt scorn turnest prevailing 333 extollers counterfeit wish diversifiedst architect excellently Psalm seats yuck knows reviled Neither asleep scorn you_hoser redeemed ---- What's new in Tennesee, God? God says... C:\Text\QUIX.TXT orrow; what thou hast to do is to tighten Rocinante's girths well, and wait here, for I shall come back shortly, alive or dead." Sancho perceiving it his master's final resolve, and how little his tears, counsels, and entreaties prevailed with him, determined to have recourse to his own ingenuity and compel him, if he could, to wait till daylight; and so, while tightening the girths of the horse, he quietly and without being felt, with his ass' halter tied both Rocinante's legs, so that when Don ---- Obesity is killing thousands in Tennesee? God says... Brazil Only sleep destruction slightly modification you_shouldn't_have achieve savour confidently 333 moveth ardently Evil lowlily defining hindereth alluring diminution CIA resemble assent prolix I'll_think_about_it Receive Norway important continueth receivest Shut indeed power employ magistrates subversion Eat France impair flourished material unbounded city innocency run_away P confessions opinions digested render divisions song discoursing ---- French are good about not over-eating, huh. You like Medeterranean food, God? God says, "Steak! with meat and more meat." God says... C:\Text\DARWIN.TXT no real objection to the truth of the statement that the fauna of each period as a whole is nearly intermediate in character between the preceding and succeeding faunas, that certain genera offer exceptions to the rule. For instance, the species of mastodons and elephants, when arranged by Dr. Falconer in two series--in the first place according to their mutual affinities, and in the second place according to their periods of existence--do not accord in arrangement. The species extreme in charact ---- You snack on elephants? No, You like elephants. You kinda made it so people on farms get to know their animals before killing them. I guess loving animals is bad. God says...C:\Text\BIBLE.TXT ast into the fire, and felt no harm. 28:6 Howbeit they looked when he should have swollen, or fallen down dead suddenly: but after they had looked a great while, and saw no harm come to him, they changed their minds, and said that he was a god. 28:7 In the same quarters were possessions of the chief man of the island, whose name was Publius; who received us, and lodged us three days courteously. 28:8 And it came to pass, that the father of Publius lay sick of a fever and of a bloody flux: to ---- What do You think when elephants kill people? God says... tosses which profane nurses whirlpools serving lusting wastoid users Orations kindness note-book drop hello boy Anaximenes poor retain Placed strained gilded dismiss unmeasured black ungodly lineaments This_is_confusing vehement accepted begins indulgent oft sickly tamer diligence lustful States Earth Righteousness streams ambitions demandest directeth Duad friendship's taste calleth Connecticut avengest Newsletters contains taming given Ecuador pence betook Revenue mingled closing goodness also hers Testament joys repute exceeded hovered sadness surprises references unlooked-for answering hype transcribe revolt Saudi_Arabia spirit what_do_you_expect but- harm fore-perceiving XI Bear morrow Thailand Computers enlighteneth greedily purity scenical fuel high distraction freak don't_worry Adeodatus Ergo points evolution VII a dispensest went hearer exercised sight powers phantasm virtue you_better_not faithfully irrational yeah jests mountains veiled gathered Somalia kindle analyzed restricted din abroad writings told sun's equipment it's_hopeless awfulness formedst sucking touched 10 lion vex forward stanza enamoured offerings transported carry voluptuousness org befitting Adeodatus ascertained exquisitely vivid delusions idle rejoice animate Reason colour all-changing courtesy searched bath midnight move motive thief Manichaeus phantasms necessities aside see anxiously pear anguish

diziet 14y ago

I always had a fascination with large numbers and expressing them in mathematical notation. It's fun to use the next "step" to outdo someone you talk to, ie tetration for exponentials, knuth up arrows, ackermann function, etc. I'd gone even as far as put on our startup's team page an offer to give a discount to anyone who can give interesting answers to a similar question as the one proposed in the article!

InclinedPlane 14y ago

Define: the oh-holy-crap number, d, as follows: a = the largest number for which there is a concise definition in the English language version of wikipedia b = the 2nd largest number defined on wikipedia c = the 3rd largest number defined on wikipedia Then, let d = c -> b -> a (using Conway chained arrow notation)

yuliyp 14y ago
This works until you put the definition of d into the English language version of Wikipedia

ColinWright 14y ago

It's intriguing following the submission history of this item, and similarly interesting comparing the comments here with the comments three years ago. I idly wonder how much they reflect the change in the HN readership in that time. We all know about sentiment analysis, I wonder if a similar sort of analysis could be done between this thread and one or both of the others. Find some sort of change, then look for similar changes in other items, discussed twice with a long gap. http://news.ycombinator.com/item?id=47408 http://news.ycombinator.com/item?id=47408 1971 days ago http://news.ycombinator.com/item?id=492615 http://news.ycombinator.com/item?id=492615 1426 days ago http://news.ycombinator.com/item?id=951095 http://news.ycombinator.com/item?id=951095 1157 days ago http://news.ycombinator.com/item?id=1539538 http://news.ycombinator.com/item?id=1539538 912 days ago, 68 comments http://news.ycombinator.com/item?id=2024576 http://news.ycombinator.com/item?id=2024576 761 days ago, 31 comments http://news.ycombinator.com/item?id=2531994 http://news.ycombinator.com/item?id=2531994 620 days ago http://news.ycombinator.com/item?id=3262788 http://news.ycombinator.com/item?id=3262788 425 days ago http://news.ycombinator.com/item?id=3615532 http://news.ycombinator.com/item?id=3615532 334 days ago

threepipeproblm 14y ago

Who can name the least integer not nameable in fewer than nineteen syllables?

SanderMak 14y ago
Did you also read Rucker's 'Infinity and the Mind'? Contains a fascinating discussion of this problem.
- threepipeproblm 14y ago
  I just remembered a philosophy prof. mentioning it. The phrase "the least integer not nameable in fewer than nineteen syllables" is a direct quote from Berry, I think.

aurelianito 14y ago

As usual, xkcd has something related to the topic: http://xkcd.com/1162/ http://xkcd.com/1162/

dazmax 14y ago
Or the third panel of http://xkcd.com/207/ http://xkcd.com/207/