The Third Coffeehouse Challenge

Here's the original problem, followed by the results. Enjoy! I have included all the responses received, which you will find interesting indeed.

A NEW Coffeehouse challenge:LIMITS OF THE POSSIBLE

A couple of Zeno's patrons were talking the other night about what is, and what is not, possible (it was late, and the expresso special of the evening was quite robust). Their line of thought was indeed in keeping with Zenoesque dialogue, it seemed, so we include it here to introduce our third Zeno's Challenge. But you be the judge as you pull up a chair, and please offer your response at the end of the scenario---and have it posted later at the Coffeehouse, if you wish.
"Suppose you were asked to write down very quickly the number '1'---say, within ten seconds," posed Maggie to her evening companion.
"No problem," retorted Charles, smirking.
"Ok," Maggie added, "And further suppose that you were asked to write down the number '2' within half the time it took you to write down the previous number---could you do that?"
"Of course," quipped Charles. "No problem. What's the big deal?"
"And how about the number '3' within half the time it took to jot down '2', and the same with '4' in half the time taken to write down '3'---could you manage that?," pushed Maggie.
"Yes, I think so," said a confident Charles, who grabbed a napkin and quickly scribbled down the four numbers well within the eighteen seconds or so time frame prescribed by the rule. "Here you are," he gestured.
"Well, Charles, I've been wondering about what would be the logical limits of an all-powerful being," added Maggie. "I mean, an omnipotent God, or any other super computational device not limited by physical constraints. Just how far could such an entity proceed with our little game? For instance, could such a being continue to produce the next hundred numbers in like fashion, with each written in one-half the time taken to write the previous one," she queried.
"Wow, that would be really fast," Charles responded, "But an all-powerful being could do it, I believe."
"Let me pose my main question, Charles," Maggie summarized. "Suppose that we use as a concrete time frame our one-hour visit here tonight at Zeno's, from 11pm to midnight, and consider the following within this framework. First, could an all-powerful being follow the rule of writing down the next number in the series in half the time taken to write down the previous number, beginning as you did with '1', to be written within ten seconds, but do this for ALL the natural numbers---in other words, by not running out of time in following the rule, as you and I would? And second, if an all-powerful being could successfully follow the rule---think of it like RUNNING A PROGRAM---wouldn't it produce all the natural numbers, and there are INFINITELY many, well within the hour's time we spent here tonight?"
"Now that's too weird!," remarked Charles. "But let me think about what would be involved---maybe in the morning---hmmm...."


From: Jason Beck
Subject: Limits of the Possible As the series progresses, the times added together will converge on 20 seconds. On the other hand, the amount of numbers that needs to be written is infinite. 20/infinity=infintesimal. If writing each number takes ANY amount of time, the counter will fail. A computer is restricted (at best) by the speed of its circuits, but God could just stop time!
From: Brian Barnes
Organization: University of Texas at Austin I just read your Zeno's challenge, and would respond in the following: to part one: an all-powerful being could do it; after all, that's why they're all powerful. to part two: yes, it would produce all the natural numbers, as if it was running a program. Not only would it do it well within an hour, it would complete the task in less than 20 (yes, twenty) seconds! Simple series calculus will verify this fact. While it would GENERATE all the numbers, one could still not VIEW all the numbers on paper -- you'd be reading forever, as there are infinitely many! This is the first time I've seen Zeno's. It's cool. --- Brian Barnes - Class of 1999 University of Texas at Austin http://uts.cc.utexas.edu/~bbarnes/ bbarnes@uts.cc.utexas.edu Dept. of Chemisty & Dept. of Philosophy
Of course he could write an infinite number of numbers in a finite time period....THere you go limiting yourself with causality again...silly humans...hes omnipotenet.....he would just cause it to happen that there would be an infinitly long list of numbers in front of him that he has just written in the allotted time...ummm...sorry about the lack o' punctuation....im lazy....damn philosophers:)...
From: Linda Hendrick
Seems to me that the argument is based on the paradigm of "time" as reality. Time is a concept conceived by man to explain the unexplainable -- even Stephen Hawking had a time of it. I would argue that there is no time except the present (hence the adage, "There's no time like the present.") Even if one could so perfectly harmonize with the universal consciousness (my way of stating an all-powerful being) that one could experience being that consciousness, one can never explain their experiences. Case in point: I can experience an orgasm (I hope), but to explain to another the sensations, feelings, and thoughts so that they understand what I mean when I use the term cannot communicate the actuality of the experience. If the person to whom I am speaking has experienced (or thinks they have experienced) an orgasm they will likely conclude they understand what I am communicating. The truth is, no one will ever know with certainty whether your experience is the same as my experience. We don't even know if it is close, although we are always making assumptions. It I am talking to someone who has never had an orgasm, they too might conclude they understand what I'm communicating, but in this case it is more likely that I will assume they don't. Words help communicate concepts, but are miserable tools for communicating experiences. The Australian aborigines (who have not had their right brains fried by an over emphasis on logic and abstract thinking) communicate experiences psychically -- a much more useful tool, but one that this culture has not cultivated. In closing, let me just say that if you want the definitive answer to Maggie's query, I would search the outback. Perhaps you will find an aborigine willing to share their experience of an all-powerful being! ciao for now, Linda
From: pomp@tsl.uu.se (Stephan Pomp)
hi, the limes of 1+1/2+1/4+....+1/2**n+... = 2, thus the time needed would be 20 seconds altogether. so under the assumtions made it would be possible to write down all the numbers.
From: Tom Page It would seem that the time alloted to write down all natural numbers would only be about 20 seconds. I suppose that an all powerful being might simply write down something like "This is the set of all natural numbers" in the space of 20 seconds, and put the rest of the hour to better use than filling up a napkin with numeric symbols.
Organization: Tilburg University As our most intellectual and heaviest pipe-fuming writer Harry Mulisch asked his nearly fundamentalist teacher: If God is omnipotent, can he make a rock he cannot lift up ? This God could produce all natural numbers (provided the ink lasts), but, with his omnipotent hand, would have to write them down in ever infinituously short periods of time - that's obvious. In a non-Windoze95 multitasking way, he'd keep the world running too... I personnally wouldn't bother checking after an hour...it takes too long too read an infinite number of numbers. Besides, an omnipotent being would not get cought in a Zenoic paradox, at least not one devised by a human being. Question: could God devise a paradox he couldn't resolve ?
Message-Id: <9510210819.AA12668@jackson.freenet.org>
X-Personal_Name: WolfMan 1st... I'm a little hazy on the conditions here: do this for ALL the natural numbers--- I believe it could make it happen for all natural nos. but for all the natural numbers...what's the last one? >>in other words, by not running out of time in following the rule Sure, if it's all powerful, time means nothing to it by our standards. 2nd... Maggie's infinite and all the natural nos. are not the same. I believe that this intelligence could perform this simple task and the deadline value would be a large one (my cpu's resolution rolled over at 1078) but if it can produce all the natural numbers, I wanta know what the last one is. So, both questions appear ambiguous to me.
From: Galileo Workstation 16
I have a couple of considerations concerning the limits of the possible: 1. If you think about the problem like Zeno's paradox, God would have plenty of time to write infinitely many numbers. If you keep halving the time interval, you only get smaller and smaller fractions of a second, but never quite reach zero; therefore, the time span would really be infinite. This idea is absurd, however, because the time period allotted would eventually have to elapse: the mathematics don't apply to the real life phenomenon of time. According to this, God couldn't do it. 2. BUT, this argument is based on the assumption that God must obey His own laws of time. If God is an all-powerful being, He can certainly "bend the rules." We may wonder how this is possible, but our tiny, finite minds can only think in terms of the laws of time set up for us by God, so it seems impossible to us, when actually it's a snap for God to do it. To illustrate this, take one of our own rules, say, "No shoes on the coffeetable." Those of us who make this rule can easily break it when we want to. It is just as easy for God to break His "time rules" if He wants to. Therefore, God really could face up to the challenge at hand!
From: UTLA54A@prodigy.com (MR GENARO R RIFFO)
It seems to me that the question itself has problems. The idea of time is a constraint on which man place on himself and other men. To speek of an ominpotent being as being constrained to time seems inherently dificult. Such a creature would not experience these problems, being that he is above the constraints of man. Of course one has to first define what is meant by that of an ominpotent God, but let us assume here that he is in the biblical sense "all knowing, without begining or end". Time would seem irelevant ot such a creature. If one were to speak of a object that could presumably do a certian action, this object itself and its ability to do a action would have to be deifne within the context of the constraints of that action.
From: SUPERVISOR@byrne.com.hk (Supervisor)
Organization: Grant Thornton Byrne Dr. Barnett, Here is my response: First, Yes & No. An all-powerful being can choose a small enough time, say X, so that it can start writing down the first number in X time and continue to do so according to the defined rule that will not exceed the given time. Using a small enough X time will approach a total of Y time at the end. And it can choose a small enough X to yield a Y that is less than the given time. However, an all- powerful being cannot write down ALL the natural numbers since there are infinitely many. Cannot write down ALL the natural numbers regardless of time. An all-powerful being can write all the natural numbers within the given time if and only if it can write down all the natural numbers. My response is NO. (Give that there are infinitely many natural numbers). Second, No. An all-powerful being cannot write down all the natural numbers. Clement Chung
Sender: root@cell.dorm.umd.edu
From: root If the all-powerful being were to follow the rules, I think that being coudl write down the entire natural number out but not within the hour time frame. Following the rule, the total time it takes looks like this: 1 10 sec 2 10 + 10/2 sec 3 10 + 10/2 + 10/4 sec
From: itsbob1@ix.netcom.com (DR. ROBERT H. KEMP )
yes!........ALL POWERFUL...... means just that....
-0400 Received: (from deniset@localhost) by picazuro.yorku.ca
X-Personal_name: denise To an all-powerful, Absolute being, all that is, has been and will be is present to its mind. Therefore, the all powerful being would not have nay trouble performing the task within the time alloted, becuase form its perspective, it would have already fi nished doing so even as you pose the task to it. .
Received: from buffalo1.localnet.com (lynx@buffalo1.localnet.com
An all powerful being would not be constrained by time, so, it could do the impossible ; ie counting all real numbers. We are the limiters because of our morality and limited thinking.
From: dw_holo@alcor.concordia.ca
X-Personal_name: Darren Holowka Subject: mailto:rbarnett@grits.valdosta.peachnet.edu Yes, an omnipotent being could do it... By definition. Omnipotent means just that- capable of ANYTHING, therefor one should not be surprised when the words omnipotent and infinity produce counter-intuitive results... after all that's what we're discussing. A better question for omnipotence is... Could God creat a stone to heavy for Him to lift?
From: Matt Peterson
Organization: Lehigh University This is the first time I have posted, and I am sure that you have heard this answer, but just in case I will give it anyway. Zeno's paradox's often rest on the belief that all numbers that can be conceptualized can be devided by 2. If we can some how prove that to be wrong, than we can solve his paradox's. There are such a thing as infinitely large numbers, numbers so large that we often use the phrase'goes to infinity'. Because there are these numbers there must be infinitly small numbers. (simpily 1 ovewr the infintly large number). 'Goes to 0' if you will or also called infanatismals. It is these numbers that can not devided by 2. They exist only in theory of course because they are so small. They can't be multiplied or devided or even written for that matter. To do so would be to violate the princepal of its smallness. But ow does this relate to the problem of time? Well when we successivly devide numbers (or time) by 2 we are in a sence going to 0. It is just between the last real number grreater than 0 and 0 that we run into these infanatismals that can not be devided by 2. Hence The omnipotent being could write his last number in that time. I look forward to your responce, you have a great page. matt
From: jennifer walsh
Well, aside from speculating on the logical abilities of an all powerful being, I believe that this couning task is impossible. Whereas the natural numbers are infinite, the amount of time used for counting these numbers(one hour) is not. So, even if the being continually divides the amount of time needed to write down each number (and the time increments become really, really short), it doesn't matter because at some point the hour will end and the being will be left with the rest of those infinite numbers that have yet to be counted. Does this seem correct to you?
From: "Mark A. Young"
Organization: Acadia University The being not limited by physical constraints could not only write down all the natural numbers within twenty seconds, but could fit them all onto the napkin by Charles's plate (by the simple expedient of writing each number in only half the space that its predecessor took). Because the being has no physical limitations, each number can be written as quickly as is necessary, and in as small a space as necessary (these being mere physical restrictions). There is no logical contradiction in the notion that the entire set of natural numbers could be written down, only the (very real) problem that being restricted by the laws of time and space could never finish the task. If you want proof that all the numbers are there, simply ask the unrestricted being to show you any one of them. By simply magnifying the appropriate part of the napkin the appropriate number of times, the required number can be brought into view. ..mark young
Sender: arturo@mesopb.obspm.fr
These are my answers to the 3rd challenge The question "could an all-powerful being follow the rule" can be reformulated as the existence of the number T=1/2^(n-1) where n is the number to be writed and T the time allowed by the rule to do it. The answer is obviously yes. T exists allways if the time is continuous, if it is quantified the answer will become negative but I think we are not discussing it ( or are we?) And to the second question the answer is again yes The powerful being could write all the numbers , no in 1 hour but in just 20 seconds. The timed wasted in the infinite number writing following the rule and with 10 sc. for writing "1" is just the solution of the sumation of the series 1,1/2,1/4,1/8,... multiplied por 10 sc. The sumation of all the infinite elements of the series gives 2 ,which multiplied by 10 gives the 20 sc. of the answer. Now, the obvious following question would be: Could the all-powerful being write no the natural numbers, but the rational numbers, allways following the same rule as before?
From: mwalden@grits.valdosta.peachnet.edu
Organization: Valdosta State University Considering that the all powerful being did not stop time to write the numbers down, it could not write all the natural numbers in a true hour. The hour of course wiuld end after many subdivisions, but however, infinity is forever, ther would be no end to the amount of numbers the being could write. Again considering the hour is indeed concrete and it did not stop time to complete it's task. Sincerely, Delma James Crosby, Music Education Major Frehman Valdosta State College.
From: sean clouse
Organization: University of Missouri - Columbia The proposition that this being is all-powerful would indeed enable him (her/it?) to write down a great many of the numbers within the allotted time. But certainly such a being would not be able to produce all of the numbers, because then that would suggest that there is a limit to the number of natural numbers. And there isn't a limit, is there? For there are infinitely many? Time would certainly run out on this being without it writing all of the numbers. The time allotted is finite; the volume of numbers is infinite. Something has to give, omnipotent or not.
From: Josh Brown
Organization: Senior, Duke University If the omnipotent being followed the rule of writing each number down in half the time taken for the previous number, and started with a ten second time frame, the infinite series would be completed at twenty seconds. At this point, I'd tap the entity on the shoulder, say, "uhhh, you forgot one," and then add 1 to whatever number he/she/it wrote last. Thus, the "God" would say, "drat" and disappear in a puff of reality. Q.E.D.
From: ArielMars@aol.com
The key word in this problem is "all-powerful being". This is an inherent use of an infinite quantity. An all-powerful being by definition can perform any formulatable task. Therefore the answer is simple: practical considerations be damned, an all-powerful being can write down any arbitrary number of numbers in any arbitrary length of time. Practical considerations being what they are, this being would eventually have to begin constructing additional universes from which to derive a fresh supply of writing material. By the end of the hour there would exist an infinite number of infinitely large universes containing infinite quantities of paper covered with numbers. Another interesting angle on the question is to note that this being would make no progress whatsoever. At any point during this hour, there still remains an infinite quantity of numbers left to be written. At all points n from 0-h the task remains to be completed: an infinite number of numbers are left to be written. Thus the all-powerful being finishes the task just as it is beginning. Infinities are fun. -Mars
From: Galileo Workstation 9 Organization: Valdosta State University Odum Library
It would seem to me that an all powerful being would indeed be able to complete the problem in the alloted time. In order to do it we must allow that this being is not restrcted in his ability or his thinking; i.e. he would already know the answer, without thinking or calculating, or would do those so fast that it would be unreasonable to lable them thinking or calculating. Also, writing would be done in a similar manner, it would just be there, or it would be done so fast as to be a reaction to the problem.
From: Bruce Galbreath
If time is continuous, then an all powerful being could write down all the natural numbers in an hour, since she would never come to an interval of time too short to be halved, and since she can move quickly enough to write any number in any finite interval of time. However, I think the puzzle of someone completing an infinite number of discrete acts of writing in an hour points to the likelihood that time is not continuous. If there are a finite number of minimal time atoms in an hour, then even an all powerful being cannot complete more actions in an hour than there are times in that hour. How do we get from one temporal point to the next? Since there are no temporal points in between, we must somehow leap. Bruce Galbreath bruceg@sover.net
From: "Jana M. Hofer"
Our omnipotent creature will have no more luck going through natural numbers, each in half the time, than the orginal arrow had in hitting the target. The arrow never reaches the target because it moves in finite space and time, and must, according the problem's formulation, move through an infinite number of conceptual points. The paradox occurs because of this confusion between real and conceptual space. Positing an omnipotent creature does not resolve the original confusion between real and conceptual space. The omnipotent creature will be no more successful than the
From: Gilles Gour
If an infinite number were a real number, even an all powerful being could not make it fit into one concrete hour, into any mumber of hours. But since an infinite number is not a real number but a number that is for ever in need of "one more" (the mathematica equivalent of human desire), even the smalest portion of time imaginable can encompass it. One consequence of this is that it would take such a small portion of time for this all powerful being to accomplish the exploit required that absolutely no real person present would notice. This would of course make it possible to keep alive the perenial discussion about the existence or non existence of such a God. Excuse my english. Gilles Gour, Montreal, Quebec.
From: Gilles Gour
If an infinite number were a real number, not even an all powerful being could make it fit in a finite amout of time. But since an infinite number is not a real number, but a number for ever in need of another "one" (a mathematical equivalent of human desire) even the smallest portion of time imaginable can encompass it. A further consequence of this is that an all powerful being could do that at an infinite speed so that nobody present could notice it. We could then go on arguing about the existence or non existence of such a God. Gilles Gour, Montreal, Quebec
From: Gilles Gour
Bonjour, My answer of a few days ago was stupid. The very conditions of the problem would prevent even an all powerful being from acomplishing the task in less than seventeen seconds an a half (the time it takes to write down the first three numbers). I am not sure I have a much better answer this time, but at least I am seriously beginning to see the difficulty. In Zeno's paradox, it takes an infinite amount of time to cross a finite distance in space because of the indefinite halfing that we know of. In that case, even an all powerful being could not do better than the tortoise. It is also true that the problem would remain the same however small the distance to cross might be. This can only mean that, however small the distance, it contains an infinite capacity to be divided in half: however small the distance, it is always bigger than a geometrical point. The difference here is time instead of space. Also, what is divided in half is not the time that is left (the rest of the hour between 11 and 12), but the amount of time taken to accomplish the preceeding task. I will stop her for the moment and write to you again if my reflection goes any further. Thanks for the game, Gilles
From: Gilles Gour
Bonjour, Third and last contribution to the solution of your riddle. I think that since, contrary to the classical Zeno's paradox, where the halving takes place INSIDE a given continuum, the halving here takes place at each step outside the time continuum, the all powerful being will never be able to make the infinite collection of natural number fit inside the one hour span. It should go on for ever, contrary to my first assumption. I still am not sure that I understand this problem fully. I can't wait to see the solution... Gilles Gour, Montreal.
From: Hays Hitzing
A- An omnipotent being, by definition, can do anything and is not constrained by any logical laws B- If an omnipotent being wrote each number in half the time it took to write the number before it, and only had an hour to do it, then that being would write however many numbers he/she could in that hour. Of course in this problem we are constraining an omnipotent being to a linear definition of time and space (which kind of takes the fun out of being omnipotent)
From: Gilles Gour
Bonjour, This is my third try at it. I am beginning to see the light. My first two answers were wrong. I think this one is right. I was fooled by the impression that this was an essentially different version of Zeno's paradox. But it is not. It is an illusion to think that since the division occurs OUTSIDE the preceding time span, that this is inherently different than the classical paradox where the hare or the tortoise or Achille himself is forever prisonner INSIDE a finite distance. Even the god here is prisonner of the first ten seconds it takes to write the first number. It is mathematicaly impossible for him to take more than twenty seconds to write down the whole list (if we can refer to it in that way) of natural numbers. The fact that the following time span is outside the preceeding one does not alter the fact that it is always the preceeding moment that is cut in half. The sum of all these fraction will never equal the first ten seconds it takes to write down the first number. If we think of that all powerful being as a real person (but he is in reality a mathematical abstraction), what happens to him is somewhat flabergasting. After twenty seconds, the rest of us in the coffee house are still there drinking and chatting but the god has left us (as they always do) and is forever prisonner of that past (for us) twenty seconds. He has in a certain manner fallen through a time crack into the trap of an inner infinity. From our point of view, time has stopped for him. He is forerver trapped in a kind of vibrating eternity.We can think that each an every instant is home to such a counting god. That's why I suspect that mathematicians are religious people.
X-Personal_name: Emanuele Pucciarelli
From: wiz@interlandsrl.it Subject: My opinion for Zeno's quest :-) Hi Ron, ...IMHO, our "God" will be over in exactly 20 seconds. Thank you by a high school student :-)) (who has to study philosophy too, of course) Emanuele