The Grim Reaper Paradox

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 (non-inclusive), 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 Zeno-type 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 space-time boundary, then it will be a non-classical, quantum gravity region, in which case it will be the beginning of time. For if there is such a non-classical region, then it is not past-eternal 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 [hep-th] 19NOV2013 3. Aron C. Wall, "The Generalized Second Law implies a Quantum Singularity Theorem," 13 Aug 2013, http://arxiv.org...