The Grim Reaper Paradox
Voting Style:  Open  Point System:  7 Point  
Started:  2/27/2014  Category:  Philosophy  
Updated:  6 years ago  Status:  Post Voting Period  
Viewed:  5,157 times  Debate No:  46870 
If there were an actually infinite number of Grim Reapers tasked with your death at a uniform interval from 1pm to 2pm, and the first Reaper strikes at 1:30, the second at 1:15, and so on... then you can't both live or die in that hour interval. But if this is the case, then an infinite set of time leads to a contradiction. Thus any bounded interval of time contains only finitely many moments. And therefore time is finite. Sources Pruss, Craig, Koons et al I accept the debate, since somebody needs to get theuir hands dirty! Pro has 2 main contentions 1.) The Grim Reaper paradox yields a contradiction and 2.) Because of 1, time cannot be infinite Since this debate is a relatively long format, I will allow Pro to expand on his opening arguments and lay any additional contentions he would like to assert/defend. I will defend the contention that infinites are indeed possible, and if they were impossible, it would not be for reasons such as these Zenotype paradoxes. It seem's Pro's contention is limited primarily to the fact time cannot be infinitely divisible, please clarify if this is your main position 

I thank Con for entertaining the merit of this argument with me. I also thank the reviewers and referees for their respective opinion and engagement on the matter irrespective of the outcome. The contentions stated above by Con are indeed what the Grim Reaper Paradox seeks to show, and so those of course will be my contentions.
The Grim Reaper Paradox A Proof by Contradiction or Indirect Proof for the Finitude of the Past The Set Up Let us say that there were an infinite number of Grim Reapers each of whom had two tasks:
1) to check whether Con is still alive at the Grim Reaper’s appointed time 2) if he is still alive, to kill him instantaneously The last Reaper (Reaper 1) performs this dual task at exactly 1:01pm. The nexttolast Reaper (Reaper 2) performs the task at exactly 1:00:30 (a half minute after 1pm) ... and so on; such that each Reaper n, is assigned the moment 1/2^n minute after 1pm. Necessary Consequences There is no first Reaper because for each Reaper n, there are infinitely many Reapers who are assigned moments of time earlier than Reaper n’s appointment. It is certain that Con does not survive because in order to live he must still be alive at one minute after one. But we have said that if he survives up until 1:01pm then Reaper 1 will strike. But Con will also not survive until 1:01pm because in order to do so he must be alive at 30 seconds after 1pm, in which case Reaper 2 will have killed him. Likewise, Con can't survive until 1/2^n minutes after 1pm, for every n. The Result Hence no Grim Reaper can have the opportunity to kill Con. So it's impossible that he survive, and also impossible that any Reaper kill him! However, it seems also to be impossible for Con to die with certainty and yet to do so without any cause. An Alternative Set Up Sans Causality But now let's do away with causal notions, welcome to Grim Space: where a Grim Reaper can both issue and fail to issue a death warrant, which is a contradiction. Consequently, the Grim place assumes that an actual infinite amount of time can exist. Say the high ranking Grim Reaper known as Vernich issues a death warrant by placing a particle the size of a point in a predetermined position, exactly the distance = d/2^n meters from a plane. (That is, arbitrarily close to the plane). But suppose there is also a Grim Mover (Vernich's servant) that checks to see if any earlier Reaper has issued a warrant by seeing if a particle is already at a distance of d/2^i meters from the plane for some i greater than n. (That is, closer to the plane than what would otherwise allow Vernich issue a death warrant). [as the exponent i in the denominator increases, the distance between the point particle and the place ultimately get infinitely close] If a particle has already been placed in the predetermined position, Vernich does nothing. But if there is no particle in an appropriate location, then Vernich issues his warrant, placing a particle exactly d/2^n meters from the plane. Necessary Consequences At 1:01pm some particle is at d meters of the plane and no particle is located there. If there is no particle at any location d/2^i meters from the plane, for any i, and there were no particle d/4 meters from the plane, then Reaper 1 would place a particle in d/2 meters. The Result Therefore a contradiction occurs. For there must be, at 1:01pm, some particle in an appropriate position, but if the particle is located at that time in position d/2^n meters from the plane, then every Grim Reaper whose number is greater than n did nothing, contrary to our hypothesis. Thus we have an impossibility associated with temporal infinity, and so time is finite in its existence. [1] Sources 1. RobKoons.net/media/83c9b25c56d629ffffff810fffffd524.pdf Addressing the Grim Reaper Paradox:
The Short Answer: The first Grim Reaper arrives and Con is dead at 1.00 pm, exactly, it’s impossible to tell exactly when Con has died or time is quantified but still infinite.
The Long Answer:
Death At exactly 1.00 pm: What Pro has provided us with is a convergent series. I am sure he has heard of this before and will remain unsatisfied, but I will provide this to set the ground in the current resolution of these types of paradoxes.
A reaper arrives at 30, 15, 7.5, 3.75… second intervals in preceding order. This is formally known as a convergent series (http://mathworld.wolfram.com...). A series of such will yield a finite answer
This can be expressed in the following series of fractions:
60/2, 60/4, 60/8, 60/16, and so on. Each Grim Reaper n+1 has 60/2^(n) time to itself before the next reaper arrives. The time the ‘first’ reaper arrives is the last term in this infinite set. Of course this is a mathematical fallacy, but one can take the complete sum and see what the sum (x) of the infite set is by the following subtraction.
1.) x = 60/2 + 60/4 + 60/8 + 60/16… 2.) 2x =60 + 60/2 + 60/4 + 60/8 3.) 2x – x = 60 + 60/2 – 60/2 + 60/4 – 60/4 + 60/8 60/8… Here all the terms after 60 cancel each other ad infinitum 4.) x = 60
Here we can see the sum of the infinite set tends to 60, or the full 1 minute. Therefore, the first reaper arrives exactly at 1.00 pm (1.01 – 0.01). A further justification can be made:
If we take 60 seconds, and subtract an infinitesimally small fraction from it, we yield 59.99*. 0.99* is known to be equal to 1, and is trivially (and not so trivially) proven by the simple:
1.) 1/3 = 0.33 recurring 2.) 0.33 recurring x 3 = 0.99 recurring. 3.) 1/3 x 3 = 1 4.) 0.99 recurring = 1
This amongst other angles of resolution yield an answer of 1.00 pm being the time of death. From this reasoning, the question of “which Grim Reaper?” becomes a meaningless question, because there is nothing infinite number of grim reapers could arriving simultaneously at 1.00 pm.
Simultaneous Causality:
This of course defies intuition, however Pro stated that the job of a Grim Reaper is to check the status of the victim before killing the victim. This implied two temporally separate events. In which case we need to either accept that these events can occur simultaneously, or the victim dies at 1.00 pm + δt, where δt is an infinitesimally small period of time. In either case, the question of asking which reaper issues the death warrant when multiple/infinite grim reapers could have done so simultaniously can be answered with “The one(s) that arrive at 1.00 pm”
Implications of the ‘Quantification’ of time: Each unit of time is not infinitely divisible
The classical notion of time is continuous, uniform and discreet. Causes preceed effects, effects take time to execute. When time is divided beyond the Planck Time (http://en.wikipedia.org...), our current understanding of physics breaks down, along with all intuitive notions on how objects that would behave within this timescale. A theory of Quantum Gravity is required in order to resolve events that occur within this timescale. Now, if one assumes time is quantified into discreet ‘jumps’, then the problem of Grim Reaper’s paradox is immediately resolved. However that does not rule out an infinite past as Pro asserts, it only rules that the number of events that occur within a discreet amount of time must be finite.
This does not rule out the past and future being finite. If there was no beginning, then there is no issue with an eternal past, and not with an eternal future.
Therefore, even if I were to concede that discreet units of time are not infinitely divisible, it would not prove Pro’s contention that time cannot be infinite, such examples are modelled in cyclic models of the universe (http://en.wikipedia.org...) amongst many.
I will address Pro’s Sans Causality scenario in the next round 

I. General Rebuttal Con's solution won't do for two reasons. First, he broke the rules and extended the argument beyond the argument. Note that the paradox clearly states that "each Reaper n, is assigned the moment 1/2^n minute after 1pm." It doesn't say at 1pm. (This rebuttal alone is enough to undercut the entirety of Con's case, but now that I have room, let's explore the matter further, you'll see that Con actually supports the paradox and gives me a chance to further elucidate its force.) Note further the graph I showed  follow the curve from left to right and you'll see that as n increases, the curve approaches infinitely close to y=0 without ever reaching it. It's the same idea here with the paradox. For there will be an infinite number of reapers set to slay Con without ever slaying him, even though his death is certain. Second, Con equivocates epistemic impossibility with metaphysical or logical possibility. He says that “it’s impossible to tell exactly when Con has died or time is quantified but still infinite.” But the question isn't whether or not we can tell if Con is dead past 1pm or not, rather the question is whether he will be dead past 1pm, given an infinity of reapers assigned uniformly to each interval of time between after 1 and before 2. Nor does the paradox depend on our epistemic ability to quantify time. Third, that because the sum of an infinite set tends or smooths towards an exact time when Con will die just is to admit the point of the graph! In an infinity, there are just as many cancellations as there are instantiations. Hence we're left back with the problem an infinite set of reapers assigned all over. Also even if those cancellations went on into actual infinity. You would have supersets of the infinite set onwards towards actual infinity. For example, if the set of all natural numbers in a power set is infinite then its a bigger infinity than the natural numbers. Take the power set of that and you get another infinity bigger than either of the previous two. I think you see where this is going, for in math you can always take power sets; if you've got one infinite set then you've got infinitely many of them all stretching out: a growing infinity of infinities such that whatever cancellation takes place, you can always take the power sets and the power sets of those power sets, all of which are an infinite set. In fact Con's objection, far from disconfirming the paradox actually further entrenches it. For it is logically impossible to both have an infinite amount of cancellations AND instantiations in the same operation. II. Simultaneous Causality This isn't germane to the argument but I'll bite anyway. There's nothing incoherent about an effect being simultaneous with the cause. For there seems to be a logically possible world in which all the bullets of 12 marksmen gods striking the death blow to the victim all at once (I’ll simultaneously anticipate Con’s rejoinder: note how this set up differs from the reaper set up, for we assigned each reaper uniformly temporal, hence this isn’t like the marksman set up). Alternatively a bowling ball can be resting on a sofa in a timeless state and the effect of the ball, the depression, would be simultaneous with the cause; the weight of the ball. Furthermore if God created time from a timeless state then God's free act of creation would necessarily be simultaneous with the effect, namely time coming to be! Therefore causes don't have to always precede their effects because there's nothing incoherent about the possible cases above: Marksmen, bowling ball, God. Either way, because there is an infinite amount of time between 1 and 2 (noninclusive), if time is actually infinite, then the duration between each reaper's dual purpose is irrelevant. Even this entrenches the argument further! III. Potentially or Actually Infinitely Divisible Units? Now what's so striking about Con's final point is that this is an implicit admission of defeat. For not only is the classical notion of time is a finite one given inflation and thermodynamics as well as relativity, but it also affirms Aristotle's answer to the Zenotype paradoxes! Namely that time not actually infinite but is potentially infinite! That is the only way that each unit of time is not infinitely divisible but discrete. But since actual infinity is what the grim paradox seeks to disprove, not the potential infinitude, then Con admits here the force of the argument.
IV. Planck Time Now scientific theories are less strong than the math under which they are built, but not only does the discreteness of time under a time slice of 10^43 seconds answer with a potential and not actual infinity assumption, but it is also the case that scientifically the early Planck era itself requires a beginning given both expansion and thermodynamics. First for expansion, if there was something on the other side of the classical spacetime boundary, then it will be a nonclassical, quantum gravity region, in which case it will be the beginning of time. For if there is such a nonclassical region, then it is not pasteternal in the classical sense, but neither can it exist literally timelessly akin to the way in which philosophers consider things like abstract objects to be timeless, or theologians take God to be timeless, for this region is in a state of constant flux and instability. Now given the indiscernibility of identicals, this is sufficient for time, so even if time, as defined in classical physics, doesn't exist in such an era, some sort of time would! But if the quantum gravity era is temporal, it can't be extended infinitely in time. For such a quantum state is not stable, and so it either produced a cosmos from an eternity past or not at all. Anthony Aguirre and John Kehayias say that "It is very difficult to devise a system  especially a quantum one  that does nothing 'forever,' then evolves! A truly stationary or periodic state which would last forever, would never evolve, whereas one with any instability will not endure for an indefinite time." 2 Hence the quantum era would itself have to have a beginning in order to explain why it transitioned just some 13Ga into classical time and space. Thus whether at the boundary or at the quantum gravity regime, time probably began to exist. Thermodynamically, a singularity theorem formulated by Aaron Waal, showed this past summer that given the validity of the second law of thermodynamics in quantum cosmology, time must have begun to exist. Unless one postulates a reversal of the arrow of time at some point in the finite past, which involves a thermodynamic beginning in time, which would raise the same sort of philosophical questions that any other sort of beginning in time would. 3 It is also interesting to note that expansion spoken of above also rules out beginningless cyclical models. For those models assume inflation all the same. Hence even cosmologically time is not actually infinite. But leave that aside, although scientists may unknowingly use "infinity" to mean potential infinity since they are rarely ever thinking of actual infinity, nevertheless anything spoken of in science itself necessarily hinges on whether it is mathematically possible that an actually infinite exists or not. So Con is guilty here of appealing to a less basic science to undercut a more basic science, he's got the pyramid of understanding flipped! And what's worse is he flips it in vain, for the two theorems I gave above from expansion and thermodynamics judo reverses his cosmogonic contentions.
Conclusion So to be clear, I agree with Con that if "time is quantified into discrete ‘jumps’, then the problem of Grim Reaper’s paradox is immediately resolved." Yes. But it is resolved assuming a potential infinity, not an actual infinity, which is what the paradox is supposed to prove  namely that only a potential infinity can exist... I'm so glad to see that Con affirms this so fast in our exchange! So of course a reality in which only a potential infinitude is possible doesn't rule out a potentially infinite future, it does however rule out an actually infinite past. By Con’s own stripes he must therefore admit that time is finite because he appealed to a Planck time unwittingly; which supposes the thermodynamics spoken of above.
Sources 2. Anthony Aguirre and John Kehayias, "Quantum Instability of the Emergent Universe, ArXiv:1306.3232v2 [hepth] 19NOV2013 3. Aron C. Wall, "The Generalized Second Law implies a Quantum Singularity Theorem," 13 Aug 2013, http://arxiv.org...
Notes on my position: I am defending 2 potential positions that will refute Pro's contention that an actual infinite in time cannot exist. These 2 positions may well be mutually exclusive, or coherent and concede some of Pro's points but the thesis of the arguments are to negate the general themes of Pro. Time might well be quantised (I.e. A certain length if time cannot be infinitely divided), which seems to be a contention that Pro is pushing, but my argument continued that that alone, even if true, does not prove Pro's assertion time is not infinite. I will defend this further later in this rebuttal. I only aim to caste enough doubt on each of Pro's premises to make the conclusion dubious and therefore unsubstantiated. I hope that clarifies my position, which is not as Pro asserts 'An admission of defeat'. General Rebuttal: Unfortunately, Pro makes the argument that there must have been a Grim Reaper, which was the first Grim Reaper, in order for the victim to die. This is attested to in everyday experience, we all travel from A to B, even though we, theoretically, need to traverse an infinite number of increasingly small increments to reach B. (See Zeno's paradoxes for such examples). We can agree that the time the victim dies is 1.00pm + 1/2^n minutes. Now, since pro has made the argument such that the first Grim Reaper, must be the n=infinity Grim Reaper, then it follows that 1/2^infinity = zero exactly. I provided two such mathematical proofs that this must equal zero. Unless Pro can rebut this then the argument stands that the victim dies at 1.00 pm. Of course mathematicians are going to wince at my deliberate depiction of infinity as a number rather than a concept, it serves it's purpose to show what would happen if I accept the assumptions Pro makes. I made the argument that simultaneous causality/effect may be an issue here, and explained why it's not a case because by my same line of reasoning, an infinite number of Grim Reapers will come at exactly 1.00pm, and simultaneously issue the death warrant, which is compatible. This also negates Pro's other point that for every nth reaper, there will be a n+1 reaper before it. Since 1/n^2 is zero, then 1/(n+1)^2 must also equal zero. Therefore, an infinite number of Grim Reapers can/must appear at exactly 1.00 pm. Infinite Past Rebuttal: Pro has not substantiated his claim that expansion and thermodynamics require that time has a beginning. While some theories do posit the singularity of the Big Bang to be an absolute beginning, it is not required in others, for there are multiple explanations for both of these. There are a number explanations compatible with an eternal universe of the 2nd Law of Thermodynamics and Inflation. One such example is eternal inflation (Or the Multiverse), where baby universes break off from the eternally inflation (http://en.wikipedia.org...), where out universe is one of an infinite number of subuniverses that broke off from the main expansion, which itself has no beginning. The only objection I have seen from Pro regarding this is what is space expanding into. However he needs to substantiate his presupposition that space needs something to expand into at all. Until then his argument holds no weight. Pro has an interesting point regarding thermodynamics and the arrow of time. Without entropy, does time mean anything? I would argue no, it is impossible to tell which state is the past or future states without an understanding of the entropy of both systems. If the entropy of two 'snapshots' in time are identical, then it follows it's impossible to tell which is the past. The Eternal Inflation depictions of an eternal universe, depict just that. An every increasing number of universes, with ever increasing entropy. Notes on Causality: All of the descriptions of time so far assume classical notions of causality. Such notions are not well preserved in the smallest of scales (very small divisions of time, space). One example includes a number of derivations of Wheeler"Feynman absorber theory (http://www.hydrogen2oxygen.net...). These examples are valid here since Pro has not defined any of the Grim Reaper's attributes in time, space and other physical processes (necessary to make his argument work). Therefore in ever decreasing increments of time and space, effects on the quantum scale become very important, and the implications of causality at such scales is likely subject of our current understanding of such in quantum physics. Conclusion: The argument has gone rather off track (my doing), but is necessary I rebut Pro's blanket argument that time cannot be infinite. Since Pro's central thesis only attacks the divisibility of a unit of time, rather the number of possible actual units, it does not prove his main point, even if there was a true contradiction in the statement, which I have defended independently. 

Vernich forfeited this round.
I think the forfeit by Pro may be due to a misunderstanding (See Comments). Back over to Pro for his closing statement. 

Vernich forfeited this round.
I will summarise briefly: 1. I have argued that an infinite as given in Vernich's thought experiment is infinitely divisible, and can be dealt with mathematically. 2. I have argued that even if time was not infinitely divisible, it does not demonstrate than an infinity in time cannot exist, and gave supporting origin if universe models that are capable of explaining such. I am rather disappointed Con went AWOL, but thanks for the brief debate anyway. Vote Con. 
Vernich  Sswdwm  Tied  

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Vernich  Sswdwm  Tied  

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I said you should post your argument but just delay the posting of it....
If you have your rebuttal then I can add it to my argument unedited when I post it this weekend so we don't skip the round?
If not then I will forfeit this round too I guess
Thanks!
This is more of a courtesy request than anything else. Would it be possible to wait most of the 2 days before responding to my rebuttal, as I am extremely busy for the next 10 days and will only have a chance to have a good crack at responding during the weekend.
Of course you don't have to as you are in no way obliged.
Best regards
"the distinction between the 'actual' and 'potential' infinite here is not as trivial as saying that the future is not the past"
... it seems to me that the distinction isn't saying that x isn't y, rather it's saying that when it comes to quantity, whatever that quantity is, it can either be extended onward towards infinity (without ever actually reaching it) OR it can be extended such that it is infinite.
BUT, given the success of the paradox, it CANNOT be extended such that it is infinite; so any quantity can only be extended onward towards infinity (without ever actually reaching it).
>When you ask "what's potential here" I would answer (given that I'm arguing against an actual infinity), I would answer that the whole set is a potential infinity, you can keep counting down the halves of a yardstick forever (granting the technology to do so increases forever as well as your lifespan) without ever actually reaching infinity. That's the idea there.
>Now the Grim Reaper paradox shows that any infinity which is considered to be "actual infinity" is a metaphysical impossibility it encounters a logical contradiction.
>I either misspoke or you read wrong that I'm committed to a past which is infinite. I personally think time is finite in its existence, meaning I think that there was a timeless state of being.
>I am in fact affirming that the future is only potentially infinite... but not because "there will never be a point in the future at which an infinite number of future times will have become past indexed from the present" ... rather as I argue in the paradox, there's a logical contradiction given an actually infinite. So that only leaves us with potential infinity, meaning time will always be finite even if it never stops existing, whatever it is.
>If something is finite in its existence, then it doesn't exist necessarily, in which case the only other alternative is that it exist contingently, that is, it undergoes generation and possibly corruption. That's just a necessary relation between things which are finite in their existence and things which are not finite.
They have to have an infinite resource, but that is impossible, because energy is not recycled, only matter is. Something has to be making them!!
No, it's impossible because you have four highly specific conditions that can't simultaneously be satisfied. Take the mathematical analogue of the GR 'paradox.'
Take a function f: [0,1] > {0,1} from times to dead  alive states. The victim is alive at the beginning and dead at the end so, first, f(0) = 1 and f(1) = 0. Second, each GR checks and kills the victim at their appointed time so for every n in the natural numbers, if f([0,1/n)) = 1, then f([1/n,1]) = 0. Basically, if the victim is alive for all times up to 1/n  at which point the nGR checks on the victim  he is killed at 1/n and dead for all times thereafter. Third, since he's alive for all times up to the time he's killed and dead all times thereafter if f(a) = 1 for any a in the domain, then f([0,a]) = 1 and if f(b) = 0 for any b in the domain, then f[b,1]) = 0. Fourth, there is a unique GR that kills the victim  consequently, there exists some natural number c such that f([1/c,1])=0 and there exists no d > c such that f([1/d,1])=0. It should be clear that no such function exists. Suppose there is some highest c in N such that f([1/c,1]) = 0. Therefore, f(1/(c+1)) = 1 since c is the highest such c by the fourth condition. But then, by the third condition, f([0,1/(c+1)]) = 1 and consequently f([0,1/(c+1)) = 1. But then given the second condition, it follows that f([1/(c+1),1]) = 0. But c + 1 > c so c is not the greatest such c and furthermore, f(1/(c+1)) = 1 and f(1/(c+1)) = 0.
But one would hardly conclude that given that such a function is impossible that a mathematical infinity is impossible  only that there can be no scenario satisfying all four specific conditions set out by the scenario. How do you go from 'there exists no function satisfying the above' to 'actual infinities cannot exist?'