Friday, January 25, 2008

The Grim Reaper Paradox

Here is a version of the Grim Reaper paradox. Say that a Grim Reaper is a being that has the following properties: It wakes up at a time between 8 and 9 am, both exclusive, and if you're alive, it instantaneously kills you, and if you're not alive, it doesn't do anything.[note 1] Suppose there are countably infinitely many Grim Reapers, and before they go to bed for the night, each sets his alarm for a time (not necessarily the same time as the other Reapers) strictly between 8 and 9 am. Suppose, also, that no other kind of death is available for you, and that you're not going to be resurrected that day.

Then, you're going to be dead at 9 am, since as long as at least one Grim Reaper wakes up during that time period, you're guaranteed to be dead. Now whether there is a paradox here depends on how the Grim Reapers individually set their alarm clocks. Suppose now that they set them in such a way that the following proposition p is true:

(p) for every time t later than 8 am, at least one of the Grim Reapers woke up strictly between 8 am and t.
Here's a useful Theorem: If the Grim Reapers choose their alarm clock times independently and uniformly over the 8-9 am interval, then P(p)=1.

Now, if p is true, then no Grim Reaper kills you. For suppose that a Grim Reaper who wakes up at some time t1, later than 8 am, kills you. If p is true, there is a Grim Reaper who woke up strictly between 8 am and t1, say at t0. But if so, then you're going to be dead right after t0, and hence the Grim Reaper who woke up at t1 is not going to do anything, since you're dead then. Hence, if p is true, no Grim Reaper kills you. On the other hand, I've shown that it is certain that a Grim Reaper kills you. Hence, if p is true, then no Grim Reaper kills you and a Grim Reaper kills you, which is absurd.

The above argument shows that some arrangements of Grim Reaper alarm clock times, namely the ones that make p be true, are impossible, because they result in your being dead and not dead at the same time. But no such objection can be made to other arrangements of Grim Reaper alarm clock times. For instance, if Grim Reaper 177 wakes up at 8:05 am, and all the other Grim Reapers happen to wake up later, there is no difficulty--Number 177 kills you, and you're dead at 9 am.

Now we have a trilemma. Either all mathematical combinations of Grim Reaper alarm clock times strictly between 8 and 9 am are possible in the above story, or some but not all, or none (in the last case, the story above is impossible whatever the times are). The hypothesis that some but not all are possible seems unlikely. Look: it's midnight, say, and we have all of these Grim Reapers setting their alarm clocks. It would be really, really odd if they were somehow compelled by the metaphysics of the situation to set their times in one of the privileged ways, unless it turns out that there are only finitely many moments of time between 8 and 9 am, so that p cannot be true. (Indeed, by the Theorem given above, these privileged ways of setting times are very unlikely if the Reapers are choosing independently, assuming that all real-numbered times between 8 and 9 am exist, which the Theorem assumes.) That leaves two hypotheses: That all the combinations are possible or none. If all the combinations are possible, so will be the ones that make p true (e.g., Reaper 1 waking up at 8:30:00, Reaper 2 at 8:15:30, Reaper 3 at 8:07:30, Reaper 4 at 8:03:45, and so on). And that's not possible.

So either there are only finitely moments of time between 8 and 9 am, or no combination of Grim Reaper alarm clock settings is possible. In the latter case, it basically follows that it's just impossible to have infinitely many Grim Reapers, whether their wakeup times are arranged so as to result in a paradox or not. So why can't there be infinitely many Grim Reapers? It seems that the only reason to suppose there can't be infinitely many Grim Reapers, even in cases where no paradox is generated, is if one thinks there can't be an actual infinity of objects in existence. And if there can't be an actual infinity of objects in existence, then there can't be an actual infinity of times in the past, since if there were an actual infinity of times, surely a new object could come into existence at each of those times.

So either there are only finitely moments of time between 8 and 9 am, or there are only finitely moments of time in the past. But if there are only finitely many moments of time in the past, there were only finitely many moments of time yesterday between 8 and 9 am, and today is no different. So in either case, a bounded interval of times contains only finitely many moments.

I am not fully convinced by this argument, but I don't have a very good response.

[This post is revised. I am grateful to Bill Craig for pointing out some sloppiness in the original.]

125 comments:

Anonymous said...

Reading this blog is like going to a mental amusement park. It must be really fun to be one of your students.

I suspect that time is, in fact, discrete. After all the physical universe in general seems to be quantum (that is, discrete), so why not time as well? IOW perhaps there are in fact no actual infinities (other than God).

Mike Almeida said...

Hence, if p is true, then no Grim Reaper kills you and a Grim Reaper kills you, which is absurd.

I think what follows is that there is no Grim reaper kills you and you are such that some Grim reaper or other kills you. I'm not sure that's absurd to believe, since it involves an infinite disjunction. If you take the infinite disjunction of propositions of the form 'Reaper n killed a', that disjunction has to be true. But, since the disjunction is infinitely long, you know that no particular disjunct true. It would be nice to have a truth-operator in this context which took wide scope on the disjunction, and narrow scope on each disjunct. Similar expressions are unproblematic in other modal contexts. Ask whether some Reaper was a wrongdoer. W(A v B) (someone or other did wrong) and ~WA & ~WB (no one in particular was a wrongdoer) are perfecly consistent.

Alexander R Pruss said...

Mike:

"there is no Grim reaper kills you and you are such that some Grim reaper or other kills you"

Putting this in terms of quantifiers, I get:

(for all g)(GR(g)->g does not kill you) and you are such that (for some g)(GR(g) and g does kill you).

This seems an explicit contradiction.

(Here, -> is the material conditional, and GR is the predicate that one is a grim reaper.)

Mike Almeida said...

Alex,

There are models for infinite choices on which both (1) and (2) are true. No theorem that would generate contradiction is valid in those models.

1. ~Mp1 & ~Mp2 & . . .& ~Mpoo

2. N(p1 v p2 v . .v poo)

Here we have it is not possible that Reaper1 kills you and not possible that Reaper2 kills you and. . .and not possible that Reaper00 kills you. But necessarily Reaper1 or . .Reaper00 kills you.

Alexander R Pruss said...

Does ~Mp entail ~p does Np entail p on these models?

Alexander R Pruss said...

Sorry, a word dropped out: Does ~Mp entail ~p and does Np entail p on these models?

Mike Almeida said...

Does ~Mp entail ~p does Np entail p on these models?

Give 'N' the interpretation 'is obligatory that; and 'M' the interpretation 'is permissible that'. Let the worlds increase in value infinitely, w1, w2, w2, . . .w00. Let 'Np' be true just in case some world where p is true is better than any world where ~p is true. Finally, let p1 = God actualizes w1. In that case, both (1) and (2) of jan 25, 12:58, are true. I haven't taken the time to work it out, but I'll be those interpretations are also right for the gr puzzle, too. Let me try to show that.

Jeremy Pierce said...

John Hawthorne thinks the mereological sum of all the reapers is what kills you.

Alexander R Pruss said...

Graham Oppy says this, too.

But how can the mereological sum kill you if none of the reapers ever swings a scythe, as in my setting? Remember my setup: "if you're alive, it instantaneously kills you, and if you're not alive, it doesn't do anything". In the case where the reaper wakeup times have a limit point at 8 am, no reaper does anything. But if no individual reaper does anything, neither does the mereological sum do anything.

Unknown said...

Assuming continuous (nondiscrete) time.

If there were 10 reapers spaced out across the hour, the reaper that arrived at 8am + 1hr/10, i.e. 8:06.

In general the reaper that arrives at 8:00 plus 1 hour/n kills. Where n is the number of reapers.

Since we have an infinite number of reapers here, n= infinity, hence the reaper that arrives at lim( n-> infinity) of 8 + 1/n kills.

Essentially the reaper that arrives at 8+1/infinity kills.

Effectively this would be 8, but not quite!

Mk said...

How is this fundamentally different to Zeno's Achilles-and-tortoise paradox?
- an infinite set of events (visits)
- of increasingly small duration
- contained within a finite, bounded time interval
- resulting, at the limit, in a binary event (killing Fred, or overtaking a tortoise).

Sure - the infinite series runs the other way ( in time ), and there is a cause-effect relationship between each Grim Reaper, but how does that negate the solution to Zeno's paradox?

Alexander R Pruss said...

The difference is that the Grim Reaper story sets up a contradiction: the victim is killed by a reaper but no reaper kills him.

Mach said...

Here's the situation in language easier for me to digest this (admittedly, much less fun, but maybe it will help clarify for others or expose where I may go wrong in this post):

(1) {a1, a2, ...} is a sequence in (0,1)
(1a) such that for each t>0, exists an a_n in (0,t)
(2) the smallest a_n does action X

We have

(1a) => smallest a_n does not exist. So we can't say the smallest one did something, so (1a) must be impossible for any sort of entity that undertakes an action. I think everyone would agree.

So, no particular grim reaper set his alarm earliest. Surely it's impossible, therefore, for the alarm settings (1a) to exist. What more this means, I'm much clear.

I don't understand the rejection of the idea that some but not all combinations could be admissible.

"It would be really, really odd if they were somehow compelled by the metaphysics of the situation to set their times in one of the privileged ways"

Why? By what measure is it odd? By our intuition on finite sets? How do we know that intuition holds up? This appears not to be a logical argument at all. Hence, I suggest it may be a false dichotomy.

Small further issue:

"(Indeed, by the Theorem given above, these privileged ways of setting times are very unlikely if the Reapers are choosing independently, assuming that all real-numbered times between 8 and 9 am exist, which the Theorem assumes.)"

The theorem shows that it is unlikely if the alarm times are chosen independently AND uniformly. I don't understand the why the latter assumption should necessarily be imposed and give us license to declare it very unlikely. (I don't understand why it should be excluded either, but I'm just pointing out it's an additional assumption of the theorem.)

Even if they were uniformly iid, probability 1 does not imply the event occurs. For a single clock, we have P{X =/= c} = 1 for any c in (8,9) -- so some event with probability zero must occur on every alarm setting.

Alexander R Pruss said...

"I don't understand the rejection of the idea that some but not all combinations could be admissible."

One thing I've since come to realize is that these Grim Reaper arguments are based on an Aristotelian metaphysics of localized powerful particulars. On such an Aristotelian metaphysics, the particulars have what powers they have, and should be freely recombinable in their different locations.

"Even if they were uniformly iid, probability 1 does not imply the event occurs."

Sure, but probability 1 implies that the event can occur. :-)

Unknown said...

Hi, I've been searching online for the history of reapers and I find this very rare kind of arguement I am not sure what the calculations are for. What I really want to ask though Is that by any means are these feedback you all create really coherent at all?

I dont mean to be a problem but i just wanted to say i found this somewhat amusing.

(Getting back into character)

And also I'd like to know what Is so important about 8 and 9 nine o clock.

Why are we going off into calculations?

And why did the unknown guy come in and post also that everyone would agree?

Is this real or a joke? I really dont mean to disturb you but I saw this and was wondering: If You can comment me on google plus. I would like to know. There was a comment zippy had made that said you make it like a mental amusement park.

Yes. I agree. What is so important about 8 and 9 o clock?
Is this real information you exaggerate or just what you just put up to speak a code only a few can understand?

What is the point of this? (I dont mean to be the agressive stranger or anything but..) I also was wondering,, please guys, dont hate me immediately,,but:
Why is it that zippy is the only one who is coherent on this post? Zippy said exactly how it looks and feels, I just don't understand. Zippy came and got fascinated by your writing just like i did. And when i think about it. Well...*gulp* we all could use a bit more discretion in this crazy world..hmmmm

Please,,listen to a helpful stranger,,
So Ok. Next up:

Take No offense, please,,please,,listen,,

please, please,,,listen:

You DO know that the average reader just doing research wont understand this stuff right? And i say that with the expectation that i just came here to research history: but i kind need to maths it seems. Hold on. This is the most maths i know:

Jesus was a shimigami promoter of long life: Please no offense of my math attempt: (j ws + A prmt of long life) Actually i think your maths is better..Is your maths real maths? It looks real.

Can I atleast get to be the stranger who get to hear What inspired this? ,,,,And,,,,I'm also Not sure if I sound pleasant enough,,but i thought that if you play around with the word 'reaper' like that,,I would definately want to talk to you if you don't mind.

I am not exactly a collage grad, but I just saw your post. What inspired this study sir?

Did you also watch the grudge, being human and Bleach the animation?

Impossible people always annoy and harrass you to leave your space.

Overall, i did get a real smile from this Blog. Ahhh and i analyse that if this was a joke,, you must know ALL about chronic dry eye and legal killings. You might also know what circadian rhythmic antidote take away stress, fear or anxiety.

Bro. me and you are on the same quest....Bro hug?

Please please please, Follow me on facebook family. Stawker Wilson.

Good day to you now until if you see fit to reply.

Cale said...

It seems an easy response to point out that the Grim Reapers you describe would have to violate the physical laws of the universe in order to execute the scenario you describe--no set of physical entities, whether finite or not, could make it happen. Essentially, you are describing a set of magical, essentially phantasmal entities who must have properties which are not just unusual, but physically impossible--unheard of in the totality of our experience.

It seems quite a bit more likely, then, that the proposed paradox is avoided because the existence of such a magical murderer itself is metaphysically impossible than because all infinite sets of actual things are metaphysically impossible.

Certainly, the argument does little and less to support the claim that actually infinite sets are metaphysically impossible--there are just too many viable, alternative metaphysical rules which would prevent the putative paradox. The above is just one of the more obvious and plausible ones that you didn't bother to address.

Alexander R Pruss said...

This is addressed in the forthcoming book.

Quick remark: Generally, philosophers don't want to make the laws of nature be metaphysically necessary.

Cale said...

Thanks for the reply. I commented because I was hoping you might point me to a more updated version of one of your GR arguments, or any similarly styled argument against an infinite past.

I responded to another similar argument of yours a year or so ago (the original post was made here on your blog around the same time as this one) and you suggested that you were moving away from the approach you outlined in that argument (very similar to this one) and towards a different sort of approach altogether. Should I just be waiting for your book, or can I find your more recent work in that vein elsewhere?

Also, note that my proposal is not precisely that we should consider the laws of nature to be metaphysically necessary and, somewhat more importantly, the fact that philosophers generally don't want to make them metaphysically necessary wouldn't salvage this argument from my objection even if that is exactly what I had suggested.

Alexander R Pruss said...

The book, which is coming out in late August, should be the best place to look.

hoblescotch said...

"The last Grim Reaper (Reaper 1) performs this dual task at exactly one
minute after noon.

The next-to-last Reaper, Reaper 2, is appointed to perform the task at
exactly one-half minute after noon.

In general, each Reaper number n is assigned the
moment 1/2^(n-1) minutes after noon.

There is no first Reaper: for each Reaper n, there are infinitely many Reapers who are assigned moments of time earlier than Reaper n’s
appointment." --Koons

There is a reason Koons states, "there is no first reaper", because in that moment, the alleged 'paradox' falls apart.

All this 'paradox' has accomplished is to create an 'infinity' hole by taking successively smaller and smaller chunks of time. This is basically a rehash of zeno's paradox, where Achilles is stuck because no matter how fast he goes, he can never arrive anywhere, because the space between himself and his goal is 'infinitely' far.

I would assume, after setting up the problem correctly, that one could simply take the limit as n approaches infinity (as is demonstrated in the link below), and would arrive at the trivial conclusion that the first grim reaper is the one with the kill, which is to say, the grim reaper that arrives (or wakes up) at the first allowable moment.

http://philsci-archive.pitt.edu/1197/1/Zeno_s_Paradoxes_-_A_Timely_Solution.pdf

Furthermore, aside from the weakness of the paradox used here, it is more astounding that the idea of 'infinitely small amounts of time' is being asserted as to not exist over this.

Perhaps within physical reality one may reach a limit of how small a measurable unit of time is, but within the confines of the alleged paradox, it actually demonstrates the opposite, that an infinite amount of time does exist and that still does not prohibit time from moving forward.

There are an infinite amount of rational numbers between 0 and 1, and if I am not mistaken, it is considered 'sparse' compared to the infinite number of reals that exist between 0 and 1. That does not mean you cannot put your car into reverse and get out of the driveway...

Michael Gonzalez said...

hoblescotch: The solution to Zeno is not to embrace the absurd idea that we actually traverse an infinite, which is not just impossible, it is a self-contradiction, and therefore meaningless. It is just like saying "sure you could have a married bachelor, so long as he remains a bachelor after getting married".

The solution to Zeno's paradox is to treat the infinity in question as merely potential rather than actual. (There aren't actually infinitely many sub-divisions in the world; there's just no limit to how many sub-divisions you could in principle do).

To use your comparison, the potential for sub-dividing the real numbers between 0 and 1 is infinite/limitless, but no one could ever actually complete this sub-dividing. "Never" is just the conceptual flip-side of "infinite". Think about it: if mathematics with limits actually told us what would happen when we arrive at the end of infinite iterations, differentials would always equal 0/0 and calculus would be meaningless. Fortunately, the whole task is meaningless from the start, since an infinite number of iterations can never actually be completed.

For the Grim Reapers, the paradox cannot be resolved by just saying there was a first. If there were, then, the number would be finite. Moreover, we can't just shrug our shoulders and point at math with infinite sets. This is an infinite sequence and the man cannot still be alive after the allotted time, and yet no particular Grim Reaper could ever meet the qualifications for killing him. He is both alive and dead. Paradox.

hoblescotch said...

Michael:

Before you comment anymore, please answer for me this question: the integral from 0 to infinity of 1/2^x

Michael Gonzalez said...

Hoblescotch:

I don't see the relevance, and it's been a minute, so I might have messed up a step, but I think the limit part goes to zero, and so minus the 0 part.... 1/ln(2),right?

Even if I got it wrong, what's the relevance?

hoblescotch said...

Michael, you just took an integral from 0 to infinity while simultaneously claiming:

"Think about it: if mathematics with limits actually told us what would happen when we arrive at the end of infinite iterations, differentials would always equal 0/0 and calculus would be meaningless."

Which is it?

Michael Gonzalez said...

Hoblescotch:

I already explained why the mathematical use of infinite limits is not the same as actually traversing or completing an infinite. An infinite limit is a potential infinite. Your original example of the numbers between 0 and 1 was sufficient. The function in your second example keeps diminishing so that it approaches zero. But it will never actually arrive. There is no value at which the result is actually zero. We just realize that that's what it's getting closer and closer to and so we treat it as zero. If there were a value at which it actually equalled zero, then the limit wouldn't actually be infinity, but some finite number.

The issue is not mathematical; it is conceptual. The entire use of terms like "infinity" is to distinguish that which could never be arrived at from that which could.

Besides, you haven't addressed the paradox of the Grim Reaper, which grants you the real infinity. Still the man must be dead by noon and yet none of the Reapers could have possibly killed him. There is no "infinity-eth" Reaper, just smaller and smaller finite numbers forever.

hoblescotch said...

Michael:

How many sums are in a Riemann integral?

How many times would you need to add 1/2^k to arrive at the number 2?

The fact is, mathematics deals with the infinite. It obviously makes you uncomfortable, as you cannot seem to accept that you integrated an infinite amount of sums over an infinite line, but your level of discomfort does not matter in the least.

The 'First' Grim Reaper:

8am <= t <= 9am

For any given grim reaper, the time that the grim reaper arrives, if each grim reaper arrives at 1/2^(n-1) before the next grim reaper,
where n is greater than zero and an integer:

a0: 9am
a1: a0-1/2^1 =
a2: a1-1/2^2 = a0 - 1/2^1 - 1/2^2
...
a_n+1: a_n - 1/2^(n+1)

Thus, we have a situation such at the 'first reaper' arrives at a_sub_infinity whereas the 'last reaper' arrives at a0.

This can be restated as:

a0 - the sum of 1/2^k; where k goes from 1 to infinity

The infinite sum, 1/2^k from 1 to infinity is known to converge at 1.

Thus, the time the 'first reaper' will arrive is at 9am - 1 hour. Which is 8am.

Pruss:

I present, the Grim Reaper Measurist to solve the alleged paradox.

Let's say, every grim reaper that is sent to kill someone from 8am to 9am must first check in with the Grim Reaper Measurist who makes a tiny mark on a ruler to denote the grim reaper and their place in line.

Now, after the person is dead, the measure is handed to you and you are asked to count the number of marks. How many marks are there?

As you are handed the Grim Reaper ruler, the Real Measurist happens to be passing by and states that you may as well count the Real ruler as it would contain everything on the Grim Reaper ruler and much, much more. Just then, a reaper intended to kill you swings his scythe, but you hold up the Real ruler to defend yourself and it is cut in half. Holding both pieces you begin to cry, but the Real measurist comes to comfort your by saying, both of the two pieces contain the same number of lines as when they are put together, which makes you truly howl in pain...

There are, when you look back at time, an infinite amount of infinite points that can be observed.

The paradox is in the understanding of infinity. It is not true that infinite events are continuously growing, as is implied by the paradox, it is that upon abstract inspection, one can always further subdivide divisions, ad infinitum.

Michael Gonzalez said...

Hoblescotch:

The issue is not that I'm uncomfortable with infinity; it's that you have not understood it. You use it perfectly well in mathematics, but that doesn't mean you've understood the concept. It is not some final number in a sequence which can be arrived at. If it were, we could ask what number directly preceded it. But there is no answer to that. No, infinity is a term used specifically to speak about situations in which you cannot arrive at the end. That's the very concept of infinity. There is no "infinity-eth" member of any set or sequence.

You ask "how many times would you need to add 1/2^k to arrive at the number 2", and the proper answer is "you would never arrive at 2". The limit that you are always approaching, but never actually reaching may be 2. But here's the key point: If you finally DID reach 2, what was the value of k at that point??. You can't specify an answer because there isn't one because there is no such thing as arriving at infinity. If you could specify a value for k at the point where we arrive at 2 then you and I would both say the number of sums was actually finite after all.

Likewise, you want to say that the "first" Grim Reaper was at 8am, but that just shows your conceptual confusion. How many times exactly would you have cut the intervals in half before finally arriving at 8am itself? If you say "infinity" is the number of cuttings, then what about the one right before that? Is it infinity minus 1? You know good and well that that is not defined, precisely because it makes no sense to treat infinity that way.

To your proposed "Measurist" solution: Yes, one can always further subdivide divisions, ad infinitum, but one can never finally make the infinity-eth sub-division, and so one can never arrive at the alleged 8am Reaper who kills me. That's the whole point of the paradox. I must be dead by 9:00, but none of the reapers would ever qualify to kill me.

I honestly prefer other paradoxes over this one, to show that an infinite past is impossible (like the two orbiting planets or the Counting Man), but I think this Grim Reaper one works just fine. Your problem is a conceptual confusion.

hoblescotch said...

Michael, I am not going to spiral into infinity with you and your myriad of points and talking passed me to prove your pet notions on the paradox.

I have presented my mathematical argument for the grim reaper scenario. Until you have addressed my first point, mathematically, I am not willing to discuss the matter further.

Michael Gonzalez said...

hoblescotch:

I definitely don't mean to talk past your or to promote an agenda without listening. I thought I had addressed your points, and then I attempted to explain where you seem to me to be conceptually (not mathematically) confused.

If there is a mathematical question or challenge you have, I can try to answer it (though I'm no mathematician). But, the issue dividing us is not mathematical; it's conceptual and linguistic. It has to do with the very meaning and proper deployment of the term "infinite" and associated terms. You and I would surely come to the same answers in any purely mathematical case.

May I ask: Do you not agree that the use of the term "infinite" and associated words in English is specifically for the purpose of distinguishing the situation from "finite" ("non-infinite") cases where there is an end or final member or... well, finitude of any sort?

hoblescotch said...

Michael...

"If there is a mathematical question or challenge you have, I can try to answer it (though I'm no mathematician)"

Please, go, you, yourself, learn mathematics, and at that point, return with an argument.

Michael Gonzalez said...

hoblescotch, I'm competent up to all the mathematics you've mentioned so far; I'm just acknowledging that I'm not a mathematician by trade, and that I have limits.

But this is not a complicated mathematical problem. It's as simple as asking "what is the highest natural number?" It's a conceptual confusion on your part; not a difference in our math competency.

Can you not deign to answer my simple question: If there can be an "infinity-eth" Reaper who kills me, then what was the value of the one right before him?

hoblescotch said...

What's your problem, Michael? I have repeatedly said, show up with a mathematical argument or bust.

Not saying it again.

Michael Gonzalez said...

I did give a mathematical argument: If there is a "first" Reaper at the end of the infinite series of sub-divisions, then what was the value of the Reaper one sub-division prior to him? I've asked this at least three times, and it is indeed a mathematical argument.

Good luck.

hoblescotch said...

Michael, you did not make any argument, you are asking a question with the intent of weakening my argument. So, present an actual argument.

My argument is clear. I stated that there is indeed a first grim reaper and I demonstrated such, mathematically. I welcome you to make a counter argument, mathematically.

Michael Gonzalez said...

My mathematical argument would start identical to yours, with the following ending:

Suppose that x is the value of k when the sum finally arrives at 1. It must be the case that for the set of real numbers, {R}:

x ∈ R
For any y, if y ∈ R, then y ≤ x

i.e. x is the highest real number.

Since there is no highest real number, 1/2^k never reaches one, and there is no first Grim Reaper.

QED.

It's not complex mathematics; it's basic. You confused "converging to 1" with actually arriving at it. It would be like claiming that a hyperbola actually eventually touches its asymptotes. It's just plain mistaken.

hoblescotch said...

Michael, allow me to remind you that you, at one point, offered your mathematical assistance to me...but, it is obvious that you are in need of assistance.

*****************************
"My mathematical argument would start identical to yours, with the following ending:"

Then copy paste it, like a gentleman would, seeing as you are using my work for yourself. I will paste it:

8am <= t <= 9am

For any given grim reaper, the time that the grim reaper arrives, if each grim reaper arrives at 1/2^(n-1) before the next grim reaper,
where n is greater than zero and an integer:

a0: 9am
a1: a0-1/2^1 =
a2: a1-1/2^2 = a0 - 1/2^1 - 1/2^2
...
a_n+1: a_n - 1/2^(n+1)

Thus, we have a situation such at the 'first reaper' arrives at a_sub_infinity whereas the 'last reaper' arrives at a0.

This can be restated as:

a0 - the sum of 1/2^k; where k goes from 1 to infinity

The infinite sum, 1/2^k from 1 to infinity is known to converge at 1.

Thus, the time the 'first reaper' will arrive is at 9am - 1 hour. Which is 8am
*****************************

"Suppose that x is the value of k when the sum finally arrives at 1."

"x ∈ R"

k is an element of the integers, why would we assume they would be equal or could be equal when the sum arrives at 1?

"i.e. x is the highest real number."

No such thing exists.

"Since there is no highest real number"

So, you write out something that is false, point it being false, and now you've won the argument? No.

"1/2^k never reaches one"

You mean, your mangling of mathematics was unintelligible and full of errors.

"there is no first Grim Reaper."

I have already proven there is while you have not proven anything.

Michael Gonzalez said...

First, I apologize for not pasting the beginning of the argument. I didn't mean to be rude.

Secondly, what do you mean by this: "k is an element of the integers, why would we assume they would be equal or could be equal when the sum arrives at 1?"

I don't know what you're referring to. I'm saying there must be a value, x, which k would have at the point where we finally reach the result of "1". If there is no such value, then we never arrive at 1. Do you disagree?

"So, you write out something that is false, point it being false, and now you've won the argument? No."

Have you never done proofs? One of the most common parts of a proof is to show that denying the claim in question would lead to a contradiction or obvious falsehood. I showed that denying my claim that there is no first Reaper requires you to set the final value of k as the highest real number, and you and I both agree there cannot be such a thing. Ergo, there cannot be a first Reaper.

hoblescotch said...

"One of the most common parts of a proof is to show that denying the claim in question would lead to a contradiction or obvious falsehood."

Which is not what you did at all. You merely make a claim, stated it was false, then proceeded to come to arbitrary conclusions.

Let me re-outline what I believe your argument to be:

Let's say the grim reapers each have a name, the name being the integer assigned to them, the index k, in this case.

So, the last reaper would have the name '0'. The second to last, '1'...and so on. You are asking me what is the 'name' of the 'first reaper'. Correct?

And, you are assuming, without an argument, so I assume you are taking the original Grim Reaper argument, that if no 'first reaper' exists, that time in the past cannot be infinite, correct?

1. Irrelevant. I never claimed to know exactly which reaper did the kill or the second to last kill nor the number of the reaper that made the kill; I made the claim that I can state unequivocally at what time the first reaper committed the kill, which I demonstrated, meaning there is indeed a first reaper.

2. Even if your 'argument' somehow disproved my argument had some validity to the question at hand of whether or not the past can be infinite (it doesn't, it is just the same argument which I have already addressed), I would counter with my secondary argument:

Reverse the roles. There was a clerical error such that instead of the last reaper being assigned the index of 0 the first reaper was.

So, now tell me, what is the number of the last reaper? As was stated in the original prompt, we know there is a last reaper BY DEFINITION. So, does the last reaper cease to exist simply because you do not know his name?

3. I have already stated that the alleged paradox is with the understanding of infinity. There are absolutely an infinite number of 'moments in time' in the past but only because there are an infinite number of observations of the past that can be made at any point in the future. There is no paradox.

Michael Gonzalez said...

I agree that the Grim Reaper paradox does not directly entail that there can't be an infinite past. I have other paradoxes that do entail that, but the Grim Reaper is just meant to cause a problem with infinities in general; not specifically the infinite past.

So, we do agree on something, more or less. However...


It doesn't matter if you are choosing "which Reaper" or "which time". Either way, you have not resolved the paradox.

You think you can give the exact time at which the killing occurs, but you are immediately confronted with the question of how many sub-divisions it took! And which sub-division was the one right before the one that finally got us to 8am? There are no answers to these questions because you cannot actually complete an infinite series.

Reversing the roles just creates the same paradox in the opposite direction, except that you have to call them "Revivers" who bring me back from the dead (to properly reverse the situation... otherwise the first Reaper would just kill me and there wouldn't be anymore). Still, you can never get to the end of infinite sub-divisions. If you could, it would make sense to ask how many divisions it took, and what was the time value for the penultimate sub-division.

hoblescotch said...


You: "but the Grim Reaper is just meant to cause a problem with infinities in general; not specifically the infinite past."

False.

"The contradiction lies in the fact that Fred will surely die before the end of the minute, but that also there is no Grim Reaper who will kill him. What this paradox seems to show, at least according to the finitists, is that there could not be finite duration of time (such as a minute) that is actually divided into infinitely many sub-regions."
https://useofreason.wordpress.com/2020/01/05/the-grim-reaper-paradox-and-the-original-solution-part-1/

You: "It doesn't matter if you are choosing "which Reaper" or "which time". Either way, you have not resolved the paradox."

False. According to the original problem, the issue is that the first reaper cannot be determined. I say, forget giving the names of the reapers according to the index but instead give it according to the time in which the reaper arrives. By doing so, I demonstrate there is no contradiction only a misunderstanding.

For example, take every moment forward from here. Would you agree that, at least, is infinite, regardless of whether something exists to measure it? If so, take every second from here into eternity and map it to the grim reaper problem by writing down the index name of each reaper for their payroll. Tell me when you will have filled out the grim reaper payroll. It will never happen and thusly demonstrating my initial proposition.

Upon inspection, at least in the abstract sense, there exists an infinite number of points given any continuous set that contains at least more than 1 point.

"Reversing the roles just creates the same paradox in the opposite direction"

Not in the context of looking at the problem from both ends. By looking at it from both ends, you may easily conclude that both the first and last reapers exist even while you may not know how many reapers exist in between them.

"Still, you can never get to the end of infinite sub-divisions."

Yes, you can, you take the limit. Just as Achilles cannot ever reach the end of the race because of the infinite possible divisions between here and there...that is, until he does.

Michael Gonzalez said...

1) The Reapers are just a way of giving character to the time intervals. The paradox works just fine without them. Run it with God or magic or whatever that determines I will die at the smallest possible interval between 8am and 9am. When will I die? And what was the interval prior to that?

2) If there is a final member of an infinite sequence, then there would be one right before that. So, if you can arrive at the end then what was the member right before that? You can't answer. And you keep dodging the problem, because you know it is fatal to your point.

3) Limits are not reachable. That's their whole purpose. They are like the asymptotes of a curve. Do you believe that curves eventually reach their asymptotes or no? If not, why not? They are approaching them forever exactly like how limits are approached forever. Why can't they touch at the end of infinity?

4) The Achilles problem is resolved by recognizing that the sub-divided intervals of his journey are merely a potential infinity; not an actual one. There is just the journey. A person could divide that in half, and then cut it in half again, and so on, but they would never reach the infinity-eth cut. It would always be some finite number. Failing to recognize the difference between potential and actual infinites was Zeno's problem and it is yours as well. You ask about the future, and it is not infinite either. At every time, there will always be a finite number of years from right now. It will go on and on, but it will never reach "infinity years". That's not a thing.

hoblescotch said...

You are not addressing anything I am saying, merely making more random assertions.

1) "When will I die? And what was the interval prior to that?"

Completely ignoring the original prompt. If you want to present an entirely new prompt, do so, but realize that we will not move forward from this one until it is exhausted.

Is there a last grim reaper? Yes. Have I demonstrated there is a first? Yes.

Have you demonstrated that either do not exist? No.

2) "If there is a final member of an infinite sequence, then there would be one right before that."

This is just a reiteration of the first.

3) "The Achilles problem is resolved by recognizing that the sub-divided intervals of his journey are merely a potential infinity; not an actual one. "

False, there are literally an infinite number of subdivisions that can be made not a 'potential' number--whatever that means.

P.S.

"You ask about the future, and it is not infinite either. At every time, there will always be a finite number of years from right now. It will go on and on, but it will never reach "infinity years". That's not a thing."

So, you are denying that the future is infinite, even while stating that it is infinite. NICE. Your logic is confoundingly bad.

Michael Gonzalez said...

Hoble: You seem to be demanding that I accept you have proven your point, and then nevertheless present my own argument proving the opposite. That's ludicrous. I have shown through many examples, questions, challenges, and counter-arguments that you did not prove what you set out to prove. You simply misused the limit concept and skipped from "the limit of a function (which, by definition is always approached but never reached; check any math text you like) has a definite value" to "the end of an infinite series can actually be reached and will have that value". That is an incorrect move, and so your argument fails.

My challenges, questions, and examples are designed to help you understand the problem. But, since you keep refusing to answer or address any of them (asymptotes, penultimate values, etc), we're not making any progress. So please just answer the following three simple questions, and then I promise to answer any challenge or question you like. Ok? I won't dodge, and I won't ignore anything. But, I'd bet anything that you can't answer any of these three questions without immediately showing the fault in your position:

1) If there is a final time interval at the end of an infinite series, then what was the interval directly prior to it?

2) If we sub-divide Achilles' trip into smaller and smaller finite numbers, at what point do we reach a number that is not finite?

3) If we talk about the future years from today on, at what point does the number of years stop being finite?

Please, just give it a shot. Conversations are two-way streets, and I did what you asked. I gave a mathematical argument. You may not like it, but I don't like yours either, so that doesn't mean we can't move forward here. Seriously, try to answer any of those 3 and we'll definitely make progress.

hoblescotch said...

"I have shown through many examples, questions, challenges, and counter-arguments that you did not prove what you set out to prove."

You have not demonstrated anything aside from your utter lack of logical ability.

1, 2, and 3 are all iterations of each other. No clue why you think repeating your failed argument over and over again somehow improves it. It does not.

Again, is there a last reaper? Again, if you reverse the index, do you have a first reaper?

Yes and yes.

Again, am I able to take an infinite sum? Again, does Achilles actually finish the race?

Yes and yes.

Again, is the future infinite? Again, am I able to map from the infinite future into any arbitrary bounded set of time in the past?

Yes and yes.

Again, am I able to give an exact time for when the last reaper arrives? Again, am I able to give an exact time for when the first reaper arrives?

Yes and yes.

If you care to disprove anything I've said there, I am all ears. Otherwise, I am ignoring you.

Michael Gonzalez said...

I'll disprove your First Reaper argument with an airtight logical deduction, and then maybe you'll actually answer my questions. I really didn't think this would be necessary....

A logical proof against your First Reaper argument, using your own notation:

P1. For any Reaper, there corresponds a value of n in (a_n+1: a_n - 1/2^(n+1)) starting with a_0 being 9am, a_1 being a_0-1/2^1, etc. (your claim).
P2. The Reaper who kills me does so at 8am (your claim).
P3. But there is no value of n that yields 8am from (a_n+1: a_n - 1/2^(n+1)) if we start with a_0 being 9am, a_1 being a_0-1/2^1, etc. (basic arithmetic).
C1. Therefore there is no Reaper at 8am (P1 + P3)
C2. Therefore there both is and isn't a Reaper at 8am (P2 + C1).
Contradiction.

This argument is sound, and so you must either reject P1 or P2, both of which I basically copied from your argument. Or you must show that P3 is false, which is what I've been asking you to do all along.

hoblescotch said...

I concur with P1 and P2 (I had better...)

I disagree with P3.

I do not need a literal value of n, I need only prove that the kill occurred at 8am, which I have done, twice over. First by taking the limit, and second by reversing the problem, making a_0 the index of the first reaper as opposed to the last.

Do you disagree that the kill occurs at 8am, if so, please demonstrate your argument.

So, if I am right, C1 is wrong.

As there is a reaper at 8am, whose first name I do not know (the k index), but whose last name I know(the m index, the inverse of the k index), but knowing the name is not relevant in this case as we are, again, trying to determine if the reaper exists not who the reaper is, which I admit, I do not know who the reaper is, but I have demonstrated that the reaper does in fact exist.

And, C2 is then also wrong.

I have proven there is a reaper at 8am whose last name is '0'.

P.S. I appreciate you making your argument more concrete. If I am lacking in concreteness, please point it out, and I will tighten up my arguments.

Michael Gonzalez said...

I apologize for not just doing that in the first place. I was taking things for granted in my own mind, and not spelling things out like I should.

I agree that the case is going to live or die on P3. So, let me try to work through your points about it:

1) It seems to me that the wording of P1 ("for ANY Reaper, there corresponds a value of n...") precludes any Reapers killing me for which there is no value of n.

2) You ask: "Do you disagree that the kill occurs at 8am, if so, please demonstrate your argument."

Response: C1 just is the statement that there is no Reaper at 8am. You'd need to deny P1 or P3, and so far you technically haven't. So far, you agree that no value of n will yield 8am and you agree that every Reaper has a value of n.

3) I don't want to leave the "limit" and "reversal" arguments unaddressed, though I don't think they get you out of either C1 or C2 of my argument.

Unfortunately, I can't think of much to say about "limits" and "reversal" beyond the things I've already said. Limits aren't meant to be actually reached (just like asymptotes aren't ever actually reached), and reversal needs to reverse the whole scenario, in which case you can never get to the Reaper we used to call "last".

I guess the key is this: If you accept P1 and P3, then an 8am Reaper is out of the question.

hoblescotch said...

1) We know a reaper arrives at 8am, and because it is the terminal point in a bounded time interval, we know that is the reaper responsible.

2) I believe 1 covers this.

3) A limit is indeed reached. If I sum 1/2^k, there are no amount of numbers I could add which would exceed 2, and from the other side, 3 - 1/2^k, no amount of subtractions I could make that would drop below 2. However, both one-sided limits are approaching the same limit, namely 2, as well as satisfying the other necessities for a limit to be taken.

Let us return to the original question at hand, is there a first reaper?

We are given a scenario such that 8am <= t <= 9am. That is a bounded time limit. There is an 'exact' instance where t = 8am as well as an 'exact' instance where t = 9am.

Given that each reaper can only operate within their own specified 'exact instant', we know that if a reaper is present at 8am, that is the only reaper present and therefore the killer.

Now, were the prompt 8am < t <= 9am....I wouldn't have a shot of talking about the 'exact' moment the kill occurred and I would be forced to stick to the 'reversed index' argument. However, that is not the case.

So, I accept P1 (and assert P2 as a consequence) but reject P3

Michael Gonzalez said...
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Michael Gonzalez said...

I'm sorry, but P3 is mathematically certain. Besides, you have tacitly agreed with it throughout our discussion. The only way to refute P3 is to give the value of n, or show that there must be one, which corresponds to 8am. But that would be like giving the position on a curve (or showing there must be one) where it actually touches its asymptote. It's obviously a doomed endeavor. There is no such value. This is why I kept harping on this single point; and why, I believe, you kept dodging it. It's the lynchpin of the whole Grim Reaper argument and the flaw in your proposed solution.

As for "prompts": the 8am <= t <= 9am prompt came from you; not Pruss' argument. Pruss' argument begins with: "Say that a Grim Reaper is a being that has the following properties: It wakes up at a time between 8 and 9 am, both exclusive...".

hoblescotch said...

"the 8am <= t <= 9am prompt came from you; not Pruss' argument."

Why did you wait until now to say that?

I have been right about my argument the entire time, you have been fallaciously arguing against it while ignoring that I am not arguing the problem...

Great job.

Happy I pushed you to be more concrete so we could point out that the prompt reads exclusive...

So, I would then just argue, if you want to know what time the first reaper makes the kill, you have to prove to me that the moment in time you are speaking of exists.

Michael Gonzalez said...

1) I hadn't thought about the exclusivity until you mentioned it the last time. But my argument stands regardless. Even if we consider the range inclusive for both 9am and 8am, you can't deny P1 (it's your own claim, and is clearly correct) and you can't deny P3 (it's arithemtically obvious), so you're stuck with C1. Calling the range inclusive doesn't change that.

2) The moment in time manifestly does not exist. That's the whole point! There is no smallest interval, so there is no Reaper corresponding to the "final" value of "n", and so there is no Reaper that can kill you. But, nevertheless, you obviously cannot survive to 9am. So, you both do and don't survive, and the only way out of that contradiction is to deny that there can actually be an infinite number of Reapers/intervals. That is literally the whole point of the argument.

hoblescotch said...

My opening argument was with inclusive time...

If the time is inclusive, there is no paradox, as I have proven but you are too dense to accept, even while admitting you have been arguing for a different prompt.

If the time is exclusive, there still is no paradox, there is only a fallacy in the way the problem is presented. There is no time t that satisfies t = t0. So, obviously you cannot find the reaper that makes the kill at that time because that moment in time does not exist and thus no reaper could be assigned it.

If the question is: Is infinity a number? The answer is no.

If the question is: Do infinities exist? The answer is yes.

If the question is: Do limits exist? Yes.

Your arguments are wrong against inclusive time but perfectly fine for exclusive time, and you were absolutely wrong about the existence of limits and the existence of infinities. I was wrong in my reading of the prompt, but my argument was sound given my reading of the prompt--which I clearly indicated several times.

Michael Gonzalez said...

Will you please just answer how exactly you can get out of P1 + P3 = C1? I mean really. I'm totally fine admitting I didn't notice or care about whether Pruss had stated that it was exclusive, because I make mistakes, and also because I don't think it makes one speck of difference to my argument. Whether the range is inclusive or not, P1 and P2 are your statements, P3 is basic math, and C1 follows logically from P1 and P3 together.

So, unless you've got an answer that you've been keeping hidden all this time, you've failed to make your case (even if I grant you the inclusive range that we had both mistakenly thought we had to work with).

hoblescotch said...

Michael, you have wasted my time by not reading my original argument and by arguing something else nonsensically. Simply claiming that I have failed does not make it so.

I have demonstrated with inclusive time that there is an exact time that the kill is made and there is only 1 reaper that can do it. Sorry you cannot comprehend that.

JCarver said...

I think what the argument shows is that you can't encounter the infinitieth member of an infinite set. If you can, the set isn't infinite. But that's not a problem with the concept, it's what you would expect. Suppose a reaper has an alarm set for 12am, then another for 12:30, then 12:45... The reaper at 12am kills you. But it isn't the "first" or "last" member of the infinite sequence of reapers, just "a" member. Suppose you encounter an infinite bookshelf that goes on in either direction, and you pick a book off the shelf. The book you picked isn't a particular numbered member, it's just a member. Suppose you come to a different infinite shelf, but it only goes on forever to your left. You pick the book off the shelf which only has a book to its left. Again, there's no reason to call this book the "last" or "first" book, it is enough just to be one of the books on the shelf.

VictorMollo said...

Is the paradox possibly related to the fact that the time interval is uncountably infinite but you only have countably infinite Grim Reapers?

Alexander R Pruss said...

It's an interesting suggestion, but I don't think so. Even if the time interval consisted only of the rational numbers, the paradox would remain.

jtveg said...

I'm probably missing something completely obvious, but why doesn't the first (earliest) 8:00 AM grim reaper kill you?

JCarver said...
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JCarver said...

After ruminating on the paradox, I still don't sense what the force of it is supposed to be. You can only be struck dead by reaper NEAREST to you in time, but there is no such reaper, obviously. You can't be killed by the "infinitieth" reaper, just like you can't count the LAST number in the set of natural numbers. The paradox, it seems to me, isn't caused by the concept of infinity itself, but a failure to see that some scenarios aren't actually possible if concrete, infinite sets of things exist. If the problem results from the very concept of infinity, then why does the paradox fail to manifest on the stipulation that there is a reaper with an alarm set for 8am, and then an infinite series of reapers at every hour on the hour later in time? Well, because THIS scenario doesn't ask you to do the impossible, namely identify the first and last members of an infinite series. (OF COURSE you can't, that's the point of it BEING infinite.) The so-called paradox just sets up a impossible scenario, it doesn't show that an infinite set of concrete things is itself impossible.

JCarver said...

Maybe I can illustrate the point another way. Suppose the following:

(1) An infinitely long numbered ruler exists.
(2) You bend the ruler in half and set both ends on the ground in front of you.
(3) The the inch on the bend in the ruler is infinitely distant, but is nonetheless a PARTICULAR inch.
(4) Thus, the inch on the bend has a number, but we cannot say which one! PARADOX!

No, not really. That is because this scenario can't happen if the ruler is infinity long. If it is, there is no half-way point and no terminal end opposite yours. But is that any reason to think you can't be holding the zero end? I see no reason why not.

VictorMollo said...

We have:
(p) for every time t later than 8 am, at least one of the Grim Reapers woke up strictly between 8 am and t.

The statement If the Grim Reapers choose their alarm clock times independently and uniformly over the 8-9 am interval, then P(p)=1. tells us that there is a procedure of setting alarm clocks such that (p) is true.

(p) says that for any time t later than 8 a.m., there is a Grim Reaper that woke up before t and hence you are dead at time t. This is true for any t > 8:00 a.m.

This is exactly equivalent to saying that there is no smallest element in an open interval. So you die at 8:00 a.m. plus epsilon, where epsilon is as small as you like.

Proof
We have a countably infinite number of Grim Reapers. If we map times from 8:00 a.m. to 9:00 a.m. to the real line (0,1), then the probability that one Grim Reaper sets his alarm clock outside the interval (0,€) is 1-€. The probability for n Grim Reapers is (1-€)^n, which tends to zero as n tends to infinity. The probability that at least one Grim Reaper sets his alarm clock in the interval (0,€) tends to one as the number of Grim Reapers tends to infinity.

We therefore have with probability one that you die indistinguishably close to 8:00 a.m.

No paradox.

Michael Gonzalez said...

Victor: If there's no value of n at which I die, then I don't die.

JCarver: the point isn't that there are no unproblematic scenarios with infinities, but rather that the Grim Reaper scenario should be unproblematic but is made impossible by the infinity in it.

VictorMollo said...

Michael: We are dealing with probabilities. For every value of t you die with probability 1.

Michael Gonzalez said...

Victor: Correct, if I've made it to that value. But, there is not value of t that will work, since there will always be an earlier one at which I would certainly have died first. Isn't that the whole point?

VictorMollo said...

Michael: correct. You die at a time indistinguishable from 8:00 a.m.

It isn't a paradox. It's a property of real numbers on an open interval. It is the same as asking what is the smallest real number of the open interval (0,1). It isn't zero, as zero isn't part of the interval. But it is as close to zero as you care to get.

My discourse about n was just to show that you will actually die, i.e. that the probability holds up for arbitrarily small intervals.

Alexander R Pruss said...

If you die right after 8:00 am, what kills you? Remember that with probability one, none of the Grim Reapers does anything: for if you're already dead, the Grim Reaper does nothing. So what is your cause of death?

VictorMollo said...

Alexander: With probability one, a Grim Reaper kills you. I showed that in my comment at 1:36 AM.

We have a continuum of Grim Reapers. Probability is only defined for a continuous distribution on an interval, not a point, so I don't see how your comment after "Remember..." applies.

JCarver said...

If the paradox is due to the nature of infinity itself, why is there no paradox on the stipulation that one of the reapers has an alarm set for 8am? Isn't the paradox due to the impossibility of a "first" alarm in a set of alarms set ever closer to, but never arriving at 8am, not the infinity of reapers? We could actually program a digital watch to set an alarm according to that rule, and it would just run the calculation forever, right? Isn't that what we're trying to do in our mind's eye when answering the question "which reaper kills you?"

JCarver said...

The digital watch makes me think of another scenario. Say all the reapers have digital watches, all in Bluetooth communication with each other. The watches have a program that says "Set an alarm between 8am and 9am but not 8am. If that time is already set by another watch, repeat." Nobody, not even the reapers, knows what the earliest alarm is. But is seems reasonable to say that at 8am all the alarms are set, and some reaper or other has an earliest alarm (there's no reason that can't be so). Then, you won't survive past 9am, even though there are an infinite number of set alarms, and we can't say exactly when you'll die between 8am and 9.

JCarver said...

Michael Gonzales: "the point isn't that there are no unproblematic scenarios with infinities, but rather that the Grim Reaper scenario should be unproblematic but is made impossible by the infinity in it."

I'm saying that the grim reaper scenario MUST be problematic, but that's because of the procedure chosen for setting the clocks, not the fact that there are infinite clocks and reapers.

Alexander R Pruss said...

Victor:

1. Only a first-to-wake-up grim reaper can kill you.

2. The probability that there exists a first-to-wake-up grim reaper is zero.

3. So, the probability that you are killed is zero.

Which step do you disagree with?

JCarver said...

(1) “ If the Grim Reapers choose their alarm clock times independently and uniformly over the 8-9 am interval, then P(p)=1.”

“(p) for every time t later than 8 am, at least one of the Grim Reapers woke up strictly between 8 am and t.”

But surely (p) doesn’t follow at all from (1). When all the reapers set their clocks, either the time on any individual clock marked the first alarm after 8am, or it didn’t, and there is at least one alarm that is the first one after 8am, given that every reaper has a set clock. It’s not as if they have to set times constantly approaching 8am but never reaching it because they run out of moments elsewhere. For every reaper you pick, it’s true that his alarm is set for some time or other, either 8am or later--- and that’s it.

JCarver said...

The following is a tangent:

You mention William Lane Craig, and of course his interest in this argument is to help establish that the number of past events is finite. But ANY argument of this kind only has force if we have reason to think that moments, seconds, hours, and so on are actually real features of reality in and of themselves. That idea seems obviously false, isn't it intuitive that time-talk just a mental convenience? We don't need to reference moments or segments of time to EXPLAIN anything about actual experience, they are just nice shorthand for organizing memory and making plans. None of the functions of time talk and time keeping devices require that moments and units of time actually exist. (Clocks are just a regular reference point, they don't actually measure a thing that exists called time.) It seems silly to think of the past as a collection of time-segments that have been piling up somehow, or of the future as blank segments we have yet to fly by us or something.

VictorMollo said...

Alexander: I disagree with step two. The statement "first-to-wake-up" is ill-defined. We have an open interval on the rationals. The statement "first-to-wake-up" is equivalent to "smallest rational on the open interval". There is no smallest rational on an open interval. But there is, with probability one, a rational smaller than any rational you care to name.

It is like saying that the sum of 1/2 + 1/4 + 1/8 + ... doesn't exist, because you can always add the next fraction to the sum. But the sum does exist and is equal to one.

Bill said...

Stepping aside from the debate of whether this paradox is mathematically soluble or not, I would like to question the assumptions themselves. How is this not simply a more complex version of the barber who shaves all and only those men in the village who do not shave themselves? Aren't we able to simply dispose of this paradox on the basis that the premises themselves are built inherently to be contradictory? Just as we are not committed to entertain other logically impossible or contradictory objects such as round-squares or the barber, can we not discard this similarly?

At a very basic level, if the paradox resolves around the idea that there will always be a potential earlier time that you could die then the conditions assigned to the reapers (the killing if you are alive, nothing if you're already dead) can never be satisfied (much like the barber). In plain words, the scenario seems set up for failure.

I very well may have missed a simple point with the above response. However, even if I had and this remains an apparent antimony, I am not disposed the say this paradox has implications for finitism in reality. Intuitively it seems much like how Zeno's paradox may have been seen by his contemporaries until modern solutions, e.g. Cantor. I suppose I may just be skeptical about a priori arguments which are prima facie difficult to conceive of.

Alexander R Pruss said...

Bill:

Here's my worry with these kinds of solutions. For certain settings of the Grim Reaper alarm clocks, there is no paradox. For other settings there is a paradox. But now if our solution is that there is just a contradiction in the assumptions, then there seems to be some kind of a magical "force of logic" that prevents certain settings of the alarm clocks.

Let me be a bit more explicit. If one of the Grim Reapers has an alarm clock set before all the others, then there is no contradiction. But now imagine that this Grim Reaper consider changing his clock for a later time. What stops him?

On my view, the problem is that the whole infinitely-many-having-an-impact-on-one story is impossible, regardless of how the alarm clocks are set.

VictorMollo said...

Maybe we can approach it like Schrödinger. The Grim Reapers are emissions from a radioactive isotope; the first time an emission is detected, the cat dies.

At 8:00 a lead wall is removed which shields the detector from all emitters.

Now with one emitter there is a good chance that the particle is emitted after 9:00 a.m., so the cat survives. But with each emitter that you add, the chance that the cat achieves a ripe old age diminishes.

When you have an infinite number of emitters, what is the chance that the cat survives past 8:00 a.m.?


Now I know there are number of problems with this analogy; radioactive decay is exponential, not uniform; it is correspondent with R, the reals, not Q, the rationals; it is not bounded; and it probably works completely differently because I am not a physicist.

But it serves the purpose of illustrating that, given an infinite number of tries, you can get arbitrarily close to anything.

I still don't see the paradox, or at least, like Zeno's paradoxes, it can be resolved within normal mathematics.

Alexander: The statement "If one of the Grim Reapers has an alarm clock set before all the others, then there is no contradiction." is only applicable (the "before all others" bit) for a finite set of Reapers.

Alexander R Pruss said...

Victor:

My statement works just fine for an infinite set. There is no contradiction in an infinite set of reapers as long as the victim was already dead before the other reapers got to work. If one has an alarm clock before all the others, then the victim is dead after that one, and the paradox is gone.

Another way to think about it is to imagine that each reaper has free will and can set its alarm clock ahead of time for any time.

If they happen to set their alarm clocks in such a way that there is a first alarm -- no paradox. If they happen to set them in such a way that there is no first alarm -- paradox. But what force prevents them from setting their alarm clocks in the latter way?

JCarver said...

It still seems to me that the paradox is in the fact that the reapers are supposed to make a decision on the basis of a rule that can't provide a decision given their infinite number, not the fact that they are infinite in number just as such. If not, why is there no paradox as long as the reapers don't have to do anything? In philosopher speak, let (s) be a scenario where for every (a) where (a) is an alarm set for time (t), time (t) is such that it passes at at moment (m) which coincides with (t) causing (a) to sound, and suppose further that for every (t) noted on a given (a), there is another (a) set for a time (t) exactly 1/2 the time nearer to 8am, and such that it is the only (a) set for that time (t), where the last alarm is set for 9am; then (s) is such that every (a) will sound at (t) with the passage of corresponding moment (m), and further, no (a) is the first to sound.

VictorMollo said...

Alexander:
The problem is not the infinite set. The set of natural numbers is infinite, but has a smallest element. The closed interval [0,1] contains an infinite number of rationals, but has both a smallest element, 0, and a largest, 1.

The problem as originally stated was a uniform distribution of an infinite number of Grim Reapers on an open interval. In this case there is no earliest Grim Reaper but there is, for any time t > 8:00, probability 1 that there is a Grim Reaper with an earlier alarm clock setting. No paradox, that's how open intervals work.

If your statement meant that you are changing the initial conditions such that the Grim Reapers are distributed on a closed interval, then of course you can have an earliest Grim Reaper.

Whether or not the Grim Reapers have free will is beyond my scope to answer :-)

Alexander R Pruss said...

If you think there is no paradox, which of these statements do you reject:

1. The victim can only die if killed by a Grim Reaper.
2. No Grim Reaper does anything.
3. The victim dies.

They can't all be true.

JCarver said...

What is the first moment after 8am? For any moment you pick, there is another moment that must occur at half the duration between 8am and the moment you name. So, there is no fist moment, and 9am can never arrive, because the first moment after 8am must elapse beforehand. If you say that time is discrete, then you must simply pick any duration you like, which is arbitrary.

Paradoxes are supposed to expose a flaw in thinking. The flaw here is to think that time ACTUALLY IS composed of little segments that fly by somehow. Time, like everything else, is an experience we have. It seems related to the combination of experience and memory. We don't actually have an experience of time segments that go by, and we didn't invent clocks to keep track of them, just like we didn't invent rulers because we discovered a bunch of inches inside a tree.

Thinking of time as little segments can be useful (in combination with a watch), but that doesn't mean the world really is that way.

There isn't literally a problem about what time bounds the first moment after 8am, or how all the time segments of the past could have "added up" to now. Time segments aren't real. It isn't all that enlightening, then, to try and hash out the nature of an eternal universe in terms of discrete time segments, which are arbitrary and fictitious. What they can do, though, is help you get a certain kind of picture across, when used correctly. What we mean by "from eternity past" is SORT OF like IMAGINING a number line that stretches to the horizon behind you, and no matter how far back you walk along it, the line always stretches to the horizon. That's an analogy, a picture. The picture just conveys a certain felling and idea, but nothing hinges on the idea that something like the number line actually exists.

JCarver said...

My philosophy degree taught me that some minds are like powerboats doing doughnuts in the harbor. Others are like tiny sailboats in the open ocean: they have no idea where they are, but they at least appreciate the immensity of the thing.

ARaybould said...

Alexander, you ask "If you think there is no paradox, which of these statements do you reject:

1. The victim can only die if killed by a Grim Reaper.
2. No Grim Reaper does anything.
3. The victim dies.

I reject 2, because the argument to establish it generates an unterminating sequence of times at which Fred is already dead, yet never exhausts the list of suspects, so to speak. Essentially, this view turns the table on the "there's no first reaper."

ARaybould said...


I think your argument proves something other than you suppose. if P(p) is 1 everywhere within the open interval (8,9), then it follows that there is no T1 such that the following holds: there exists T0: 8 < T0 < T1 and Fred has not been killed before T1.

Your argument looks like mathematical induction, but I do not think it is, for two reasons: firstly, it does not show that the induction step follows from the truth of the predecessor (it actually shows that the truth of the induction step would entail the falsity of its predecessor); secondly, it does not have a basis, as shown by the argument in the above paragraph. The fact that this "induction" goes backwards in time tends to hide these issues.

Alexander R Pruss said...

It's not an inductive argument, but a reductio ad absurdum. Suppose the Reaper at time t kills. Then the victim was alive at the earlier time. But if the victim at the earlier time was alive, the earlier Reaper killed him. So, the victim was killed twice. That's absurd. So, the reaper at t did not kill him.

ARaybould said...

Fair enough, but this argument depends on there being reapers activated before t, and it does not rule out one of them killing Fred. Applying the argument recursively to any of those merely results in infinite regress confined to the period after Fred was killed. At any point, there will always be infinite reapers who have not yet been ruled out.

ARaybould said...

Despite my reservations, I do think there is a paradox here, but it is not about time: it is just another paradox of infinity (or an old one in a different guise), and I think it may be resolvable.

For the sake of this discussion, I am assuming that the reapers can be identified by the rational numbers in the open interval (8, 9), as I assume was intended. I don't think there is any insuperable problem if several reapers activate at the exact same time, or if there are some finite gaps between consecutive activations, but I will not consider these issues here.

Firstly, let's note that there is no paradox if we use the closed interval [8, 9]: the victim is killed at exactly 8am, by the first reaper. In this case, there is no difficulty in identifying the first reaper, as the sequence of reapers has a well-defined beginning (defined by fiat, in fact.)

In going to the open interval (8, 9), we drop exactly two reapers. If 8 had a successor in the rational numbers, the corresponding reaper would be the assassin, but it does not. No function or algorithm can find the beginning of an open interval of rational numbers or the successor of one, as, for any candidate, there is always another rational between it and the lower number from the interval's definition (that is a rather awkward phrase, as I have to avoid saying anything implying that the interval contains that number.)

Is it right, then, to say that the interval has no beginning? I think there are some intuitions suggesting that it has a beginning, even if there is no way to identify it. After all, the sequence does not go back forever, and the greater-than and lesser-than relations are strict total orders on the rational numbers and their subsets (caveat: are open intervals subsets?) The only difference, in comparison to the closed interval, which does have a beginning, is the removal of one number from each end.

In the paradox, it seems that a) if the subject was killed in the interval, it was by the first reaper to activate, and b) the subject was killed during the interval. Is there any problem with saying  that the subject was killed by the first activated reaper, even if there is no (other) way to identify it? And if one insists that the concept of first reaper is not well-defined here (i.e. that this argument does not stand as a way to define the first reaper) then proposition a) is not well-formed, but then we can still hold b), and say that there is no open interval between 8 and the killing of the subject. Either way, the paradox is resolved, without mentioning time.

Perhaps we could say, in analogy with infinity, that there is a potential first member of an open interval, but not an actual one? In that case, this paradox is an already much-debated one in another guise.

By the way, I came to your post from Boxing Pythagoras, which is where I got the notion of calling the victim Fred in my previous posts.

Alexander R Pruss said...

I don't see how the subject could be killed by the first activated reaper in light of the argument that for every reaper R the subject was killed before R awoke.

ARaybould said...

As I see it, that argument leaves some reapers unexamined. At any iteration considering the reaper which activates at time t, there are an infinite number of reapers in the open interval (8, t) which have not been ruled out. This does not change (and does not converge on a finite number) no matter how many steps you take it to.

I think we both agree that it does not find the assassin, but I do not think we can take it as saying anything about every reaper, as it will never consider all of them. It seems to me that what this argument shows is that no reaper kills the victim after the victim is killed.

This is the same issue, I think, as in Zeno's "Achilles against the tortoise" argument. The argument that Achilles cannot pass the tortoise is constructed in such a way that it never gets to consider the time when Achilles catches up with the tortoise, or any time thereafter. All it shows is that Achilles does not pass the tortoise while he has not caught up to it.

ARaybould said...

For those of us struggling with the issue of there being no minimum of an open interval of rationals (which included me until just now, despite my attempts to put it aside), the infimum reaper did it!

Alexander R Pruss said...

Don't think about this argument in terms of iterations. I want to prove that there is no reaper that kills Fred. How do I do that? I consider an "arbitrary reaper", call him x. And I show you that x couldn't have killed Fred.

That's exactly how I prove that, say, no prime number is divisible by four. I say: Consider an "arbitrary prime", call it x. I then say: either x is even or not. If x is not even, it can't be divisible by four, since any number divisible by four is even. If x is even, then x=2, since the only even prime is 2. And 2 is not divisible by four. So, x is not divisible by four.

There are no stages or iterations, just an argument that applies directly to every case to establish the conclusion. In particular, it's not an inductive argument.

ARaybould said...

There is a non-iterative way of making my point, as well: pick an arbitrary reaper, which wakes at a time we will designate as t. Your claim is that it can not be the assassin, but for any t, there are infinite reapers in the interval (8, t), which your argument does not rule out, so you cannot say this is true for all reapers. In fact, your argument that the reaper waking at time t cannot be the assassin depends on an earlier assassin doing the deed!

Let's also look at what is being generalized here: it is that if there was a prior-waking reaper, then the reaper waking at t did not do the deed. This is true, but if, somehow, you did pick the reaper waking at the infimum of all the times reapers wake up on (I'm not saying you can do this) there was no earlier-waking reaper, so your argument does not rule out this reaper.

ARaybould said...
This comment has been removed by the author.
ARaybould said...

In my previous post, I made a mistake in the first paragraph. The issue with the reapers in the (8, t) interval is not just that they have not been ruled out, but that the argument for the reaper at t not being the assassin requires there to be a killer in this interval. Otherwise, if there is no reaper in this interval (i.e. when t is the infimum of (8, 9)), then the reaper at t is the culprit.

To be clear, t here is t1 from the article's original argument, and t0 from that argument is in the interval (8, t).

ARaybould said...

On further investigation, I found that the infimum of an open sequence of rationals is not in the sequence. If you consider the set of times when the reapers activate as a subset of the reals, however... I'm not sure yet what one can say about that. It is also possible (if I understand correctly) to form a series of rationals which converges on an irrational, and so has an irrational as its infimum, and this might be the basis for a more sophisticated Grim Reaper paradox.

Nevertheless, I still feel that the particular argument being discussed here does not demonstrate a paradox, as it seems nothing more than the claim that, for any time that is a rational number after 8 o'clock one picks, either the subject was killed by a reaper activating at that time, or was killed before. That, in itself, does not seem at all paradoxical.

ARaybould said...

I wonder if it is helpful to compare this thought experiment with one that differs only in that the reapers' wakeup times are over the closed interval [8, 9]. In this case, it is equally true, for any given reaper, that either it is itself the assassin, or the assassin woke before it. Nevertheless, in this case, we can say which reaper did the deed: it is the one waking at 8 o'clock.

So what is the difference? One thing is that using a closed interval gives us more information than we have in the open interval case - information which allows us to deduce which reaper did it (specifically, it was the first reaper, which is well-defined in the closed-interval case.)

This view was prompted by reading about Paul Benacerraf's response to the Thompson's Lamp paradox, concerning how much information is given in the problem statement.

swaggerswaggmann said...

Very simply, all reapers are a continuum, there is no first and no last, but a reaperness from 8am to 9am.
This case is as realistic as a arbitrary large number of reapers.

With a probability 1 you will be dead after 8 am. Quintillion upon quintillion of reapers hitting you at the same time, 8am.
Math proved that 0.999..99=1 so at least two reapers are the first.

Michael Gonzalez said...

A continuum won't work for the singular incident of death. In any case, causal finitism is associated with individual causes, and so the individual reapers (rather than a continuum of "reaperness") seems like an intentional and indispensable part of the argument.

ARaybould said...

Swaggerswaggmann: In some presentations of the paradox, you would be on to something by drawing an analogy to the fact that 0.999... = 1, as these versions present the times of reaper activation as a series converging on 8am. Here, however, Professor Pruss may have avoided that by distributing his reapers randomly (with uniform probability) over the interval.

Furthermore, even in those cases, it seems to me that your argument goes astray when you say "... so at least two reapers are the first" - if I am correct in assuming that the two you are referring to are one analogous to 0.999... and another analogous to 1. The point of 0.999... = 1, however, is that they are one (pun intended, if that is a pun) and the same.

Michael Gonzalez is surely correct (at least in my opinion) when he says that there is a singular death and therefore, presumably, a singular reaper causing it. Furthermore, it seems that this must have been the first (in time) reaper to act, by the very argument that is central to the exposition of the paradox, which rules out all the other reapers (more precisely, as this version does not rule out more than one reaper activating at the same time, we can only say that it was one of the first batch to wake.)

So, one reaper did the deed, and we may also have one way of identifying which one (by temporal order, but only if there is a singular first reaper). There only seems to be a paradox if we feel we should be able to say more than this about the culprit, but the paradox is framed in such a way as to rule out a couple of the usually-present options. Firstly, we cannot (with the limited information we have) assign an ordinal number to this reaper (other than by the temporal order in which they wake), even if they wake sequentially. Secondly, when it comes to what time the reaper acted, the version here may differ from those using a series converging on 8am: in those cases we can say, by essentially the argument that 0.999... = 1, that the act occurred at 8am (and so the paradox statement is inconsistent by both specifying the series and saying that the 'open season' is an open interval), but here, where all we know is that we have some distribution over an open interval, we cannot identify a specific time. We can see clearly why this is so, however, so the only puzzle is why it would seem paradoxical at all.

swaggerswaggmann said...
This comment has been removed by the author.
Michael Gonzalez said...

ARaybould: I feel you like you may be missing the point of how the paradox is framed. No one is claiming that there are no scenarios (like, say, one with closed intervals) which would be unparadoxical. The point is that, having set up the scenario as Pruss has done, it is only paradoxical because of the infinity involved. So, a perfectly coherent scenario is made incoherent solely by the presence of infinitely many causes. Given that situation, the right thing to do seems to be to deny that there can be infinitely many causes. Otherwise, we'd need to appeal to something else about the scenario that is forbidden, and there's nothing to point to.

I'm also not sure I like phrasing the matter as a lack of information. If there is no Reaper at 8am, then it's not that we lack information about the one at 8am.

swaggerswaggmann said...

Araybould, if there are infinitely many present with uniform probability, there will be some that satisfy my exemple, elseway you could add some until it does. Because if my infidecimal is not satisfied (1 reaper at 8h01, the others after 8h02) there is no paradox, as you said, the first did it.
Pruss by arguing a paradox must agree that there have to be spaced so that the paradox can exist, a continuum. Then the continuum did it.

swaggerswaggmann said...

Gonzales, it is not incoherent, simply notice that putting infinity usually complexity things. Sure it is not 3 century's bc math or philosophy, but there is no harm to notice something unexpected, like Thompson lamp, in "quantum superposition" after some time, not intuitively normal, but in line with modern discoveries.

Michael Gonzalez said...

The incoherent result is that no Grim Reaper can kill him, and yet he cannot remain alive by 9am. You cannot be killed by a continuum of Grim "Reaperness". That doesn't mean anything, and even if it did, it would violate the stipulations of the scenario. I just don't think you've understood the scenario. The man can only be killed by a Grim Reaper swinging its scythe, and no Grim Reaper will attack if a previous Reaper already did.

Michael Gonzalez said...

By the way, if something is actually meaningless, then it does not count as "something unexpected" or as a "discovery". It's nonsense. Sense (by which I mean "meaningfulness or coherence of a statement") is a precondition of truth.

swaggerswaggmann said...

He is dead at 8am, killed by the continuum,as not one in particular did it. Don't worry if you can't fathom : quantum physics is meaningless, "how can it be two different opposite things at the same time ?"
But reality doesn't seems to care about your intuitions or definitions.
The continuum lift its scythes and hit.
The scenario is like trying to do relativistic physics uning only newtonian, it doesn't means that a particle can overtake light, simply that your assumptions ( one reaper.. ) are wrong and a new phenomenon ( continuum ) resolve the problem.

But I noticed you already had problems with infinity before in this thread, so I will stop here.

Michael Gonzalez said...

You have misunderstood both the scenario and my position. Let me clarify:

1) To say he dies at 8am, or that no one in particular did it, is to violate the rules of the scenario. It's like trying to win at chess by flipping the board.

2) Quantum physics does not require contradictions to be true. Contradictions aren't either true or false; they are meaningless. If some interpretation of quantum mechanics entails a contradiction, then why wouldn't we rule that one out on the grounds that it makes no sense? Indeed, that's what Erwin Schrodinger was trying to tell us with his "cat" analogy. If you read the paper, he is explicitly saying that, since some interpretations of QM lead to thinking a cat can be both alive and dead at the same time, those interpretations should be rejected as nonsense.

3) It's not like trying to do relativistic physics using only Newtonian mechanics; it's sticking to the stipulations of the scenario. If Pruss' analogy had started with "imagine a world run by Newtonian mechanics..." and then he had spelled out some interesting or strange result, it would be silly to respond by appealing to relativistic physics! The right answer to such a person would be "didn't you read the analogy? I'm explicitly asking a question about a situation within Newtonian mechanics..."

4) The analogy in this thread shows a problem with causal infinities. It's not that I have some problem; it's that there are genuine issues with trying to instantiate an actual infinity in the world. It creates paradoxes. That's not my problem.

swaggerswaggmann said...

Ok, last one.
1 no it put reality in a tought experiment.
2 never argued for a macroscopic cat, atoms do it EMPIRICALLY read about neutron diffraction. Two ways at the same time, a isolated atom in superposition... all that is meaningless crap until reinterpreted then it is simply strange, like the continuum of reaperness.

3 it show that a unexpected result can happen even if you don't like it, I used that as a exemple of paragism change. It simply show that thoses assumption are wrong, and that a new phenomenon emerge.

4 no paradox, only misunderstanding, I read the thread most response seems valid, even if you don't like them.

Michael Gonzalez said...

Swag, I appreciate and respect your position; but, overtly violating the rules of a scenario is not an "unexpected result". It's failing to engage properly with the scenario. Within the stipulations of the scenario, the only way to avoid paradox is by embracing causal finitism. I could discuss the cases about superposition and any other example you like; but they won't change the fact that Pruss' argument has specifications that you are violating in order to get around an unwanted consequence. Flipping the board does not win the chess match.

ARaybould said...

Michael: At least for the purpose of this discussion, we may have two different types of Grim Reaper paradox: those that have the reapers waking on a schedule set by a series that converges on a specific time, and, on the other hand, the one presented here, in which the reapers wake at unspecified times within an open interval. Some of my statements referred only to one or the other of these variants.

In particular, I am not saying, in the second case (i.e. Alexander's version), that there is a reaper waking at 8am, I am saying that we do not have sufficient information to pick out the specific assassin [1], and this is not paradoxical (it is merely a deliberate consequence of how the problem is framed.) What we can say is that the assassin woke infinitesimally shortly after 8am, but there are infinite reapers fitting that description. This is no more paradoxical than the scenario where all the reapers wake at 8:10 - in both cases, we can pin down the deed to a precise time (an infinitesimal interval is a precise time), but we don't have sufficient information to pick out which one did it (and in neither case are we logically obligated to conclude they all had a hand in it.)

The other version, in which the reapers awaken to a schedule set by a series converging on 8am, may be slightly different. As an example, consider the schedule set by the sequence where T0 = 9 and  Tn, for n > 0, =  Tn-1 - 9/(10^n) [2]. In this sequence, the nth term is 9 minus the nth partial sum of the infinite series (from n=1) in which the nth term is 9/10^n. This latter series converges on 1, and its nth partial sum is 0.999... taken out to n decimal places; you will find this series in explanations for the widely (but not universally) accepted proposition that 0.999... = 1. It seems to me that if, like most mathematicians, you accept this argument for 0.999... being equal to 1, you should not have any difficulty in accepting that, in this particular version of the grim reaper paradox, the assassin woke at 9am - 0.999...hours = 9am - 1 hour = 8am!

Continued...

[1] One complication of Alexander's formulation is that it cannot rule out more than one reaper waking at exactly the same time (as he writes, "Each [reaper] sets his alarm for a time (not necessarily the same time as the other reapers)", but it is not necessarily different from the waking time of every other reaper, either; as they are setting their alarm clocks independently, unique waking times cannot be assured.) In such a case, maybe one reaper does the deed, or maybe more than one is involved - the problem seems underspecified here.

[2] Is there a way of posting math expressions here? I don't want to try one of the common markup methods and end up posting gibberish that I cannot edit. I hope spelling it out as I do here is not too difficult to read.

ARaybould said...

Returning to Alexander's version, the crux here is the argument that purports to prove the proposition that the victim was not killed by a reaper, but it explicitly depends on the victim being killed by a reaper. That's self-contradictory! What we actually have here is a proof-by-contradiction argument for the proposition that you cannot pick out the assassin, no matter what argument you try (one comment to a different post on the paradox suggested the assassin could write his wake-up time on a note placed in the victim's pocket, but that runs into the potential-versus-actual infinity problem, as it would require infinite digits to uniquely identify the assassin.)

We can see this in what Alexander himself writes: "Suppose that a Grim Reaper who wakes up at some time t1, later than 8 am, kills you. If p is true, there is a Grim Reaper who woke up strictly between 8 am and t1, say at t0. But if so, then you're going to be dead right after t0, and hence the Grim Reaper who woke up at t1 is not going to do anything, since you're dead then." So, at t1, you are dead, killed by a reaper (this argument leaves the reaper at t0 a suspect, but it neither proves nor claims that this reaper either is or is not the assassin.) As, at t1, you are dead (Alexander states this, just in case you doubted it), one cannot say that this shows no Grim reaper kills you.

As to why this seems paradoxical, I suspect that many people are thinking of it as an 'ordinary' finite-number-of-reapers problem, to which they selectively add just one of the things that are changed in these particular formulations, namely the impossibility of picking out the assassin (something you can do with finite reapers, though only if you arrange things so that there is something (such as waking time) to uniquely identify each one and assign them to a sequence.) Even that may not, in itself, be enough to make it seem paradoxical, but adding a plausible-seeming argument that (according to some people) no reaper acted, or (according to some others) somehow all of them did (when it does not actually entail either), is enough to get you there.

Michael Gonzalez said...

ARaybould:

I don't think you've understood the point of the scenario. Pruss is not here denying that other scenarios (such as ones which permit an 8am Reaper) may be non-problematic. What he's done is to craft a scenario in a very specific way, which should still be non-problematic, were it not for the problems caused by the infinity. If we accept that there can only be a finite number of intervals between 8am and 9am (i.e., if we accept Finitism), then there is no problem. But, if we deny that, then it is both the case that I am killed by a Reaper before 9am and that none of the possible Reapers could have done it.

You talk as though he is denying us crucial information; but, what he's actually doing is creating stipulations for the scenario, within which there is a trilemma that seems to only be escapable via Finitism.

Now, I personally don't think 0.999... equals 1, and I think there are certain incoherencies at a much simpler and more obvious level with trying to apply the math of infinite series or transfinite arithmetic to the real world. But, I don't think you need to agree with me on any of that in order to see the paradox in Pruss' scenario.

ARaybould said...

Michael: That is odd, for I, in turn, am getting the distinct impression that you are misunderstanding my posts - in particular, I am not claiming that the convergent-series version invalidates Pruss's version; I am simply taking a broader look at Grim Reaper paradoxes in general. In my posts, you will find plenty of attention given specifically to Pruss's scenario, without any mention of convergent series. I could pull them out and post them independently, but I don't think that will be necessary.

Furthermore, The purpose of the argument is somewhat beside the point, insofar as the validity and soundness of an argument does not change depending on what purpose it is intended to serve.

As you don't think 0.999... = 1, then, from your point of view, there may well be no distinction between versions presenting the reapers' waking on a schedule determined by a convergent series, and Pruss's formulation (though I wonder why, in that case, he did not simply go with one of the versions presented earlier.)  That, however, does nothing to dismiss or render moot the issues I have with Pruss's specific arguments; those concerns have been raised without any reference to the convergent series version (and with specific reference to Pruss's argument and the things he says about it.)

It is unhelpfully teleological to say that I "talk as though he is denying us crucial information." I am merely observing that the way the problem is set up does not provide us with sufficient information to determine which reaper did the deed.

As for whether there is a paradox here: paradoxes are subjective, so there is little to be gained by arguing whether something presents a paradox. If someone sees self-contradiction where there is none, the objective way to achieve a resolution is to explain why there is, in fact, no contradiction. It would also helpful if we could figure out why a person might think there is a contradiction, and I made an attempt at that in my previous post (which is not to say that my concerns are ungrounded if it is not actually the reason for people seeing a paradox.)

Continued...

ARaybould said...

Now let's turn to the purpose of the argument. I'm not saying either that finitism is a false premise about the world or that it would not be a resolution of this paradox, but I don't think it is the the only possible one. Furthermore, I am very skeptical that this argument proves that it is, and, more generally, I am skeptical that arguments over physically unrealizable circumstances can prove something about physical reality.

One resolution might be that this scenario is physically unrealizable [1]. For example, it requires infinite pyhsical entities to be present and doing physical work within striking distance of the victim, and it requires the scenario to be set up in finite time. I suppose, based on your previous responses, that right now you are thinking that this is "violating the rules of the scenario" (yet invoking finitism is not?) If, however, you want to draw physical conclusions from it, the burden is on you to show why its physical impossibility is not an impediment to doing so.

Another possible resolution is that modeling time analogously to the number line (which, in turn, analogizes numbers to a spatial line...) breaks down in physically unrealizable cicumstances. What is the argument for this being an inadequate response, and for finitism being the only response that works?

The only reason you have given, in your last post, for taking this as proving finitism, is that it seems there is no alternative. That falls somewhat short of a proof, and it might be a case of what Dennett calls the Philosophers' Syndrome: mistaking a lack of imagination for an insight into necessity (though unconstrained imagination has its problems, too...) 

As I said in my previous post, the crux of the matter is the claim that (unless finitism is correct) " it is both the case that I am killed by a Reaper before 9am and that none of the possible Reapers could have done it." All you are doing here is repeating that assertion without addressing the concerns I have raised against the justification for second clause of that conjunction, and until someone does, we are not making any progress.

Continued...

[1] If there are only finite intervals in an hour, that would make the scenario physically unrealizable, I think, but as that is the conclusion allegedly proven by this argument, it would be begging the question to premise it. 

ARaybould said...

I think I can now say a bit more about what seems wrong about this paradox. Let's take a look at the arguments for each side of the contradiction.

Firstly, "I am killed by a reaper before 9am." The argument for that goes like this: 1: Only a reaper can kill me before 9am (if we don't accept that premise, the argument has other problems.) 2: If I am alive when a reaper awakes, it will kill me. 3:  A reaper awakes before 9am. Conclusion: I am killed by a reaper before 9am. That seems pretty solid.

Secondly, " None of the possible reapers could have [killed me before 9am]." The argument here goes like this: 1: Only a reaper can kill me before 9am. 2: If, and only if, I am alive when a reaper awakes at time t (strictly between 8 and 9), then I will be killed by it at that time. 3: I have already been killed by a reaper by time t. Conclusion: I am not killed by the reaper waking at time t. (Note that I am following Pruss's lead in phrasing the argument in terms of a specific time (t1 in his exposition.) Extending it to all reapers was left as an exercise for the reader.)

The thing here, however, is that premise 3 of the second argument, all by itself, entails the truth of the other horn of the alleged paradox (it is, in fact, a stronger version of it.) A paradox has two independent valid arguments leading to contradictory conclusions - take, for example, the Ross-Littlewood paradox, where the argument for the vase ending up empty does not entail that it ends up containing infinitely many balls, and vice-versa. We resolve a paradox by showing that one or the other horn does not hold (or possibly neither), but here, the argument for the second horn cannot be sustained without also entailing the other (this is not so the other way round: the argument for having been killed by a reaper stands on its own.)

Furthermore, because premise 3 is the other horn of the alleged paradox, and the horns of a paradox contradict each other, this argument is self-contradictory. A self-contradictory argument does not create a paradox! It is either invalid, or it is a proof by contradiction against the premise it is nominally arguing for. 

Michael Gonzalez said...

I apologize if I've failed to understand something, but we seem to be talking past each other on multiple fronts here. Let me try and at least explain how things look from my perspective. I'm going to try and be brief, since I don't think this is the proper setting for very long responses:

First off, when I point to the purpose of Pruss' specific formulation, it is because that formulation alone, if it is only resolvable by accepting Finitism, is sufficient to warrant belief in Finitism. It does so wholly regardless of how other Grim Reaper scenarios play out.

Secondly, my point about information was that it is incorrect to talk as though what's going on is a denial or lack of information. That's not the issue. We have all the information that would suffice in any finite case. So, again, the problems only start if we introduce infinity.

Now, to address the two key points that I see as hugely mistaken in your most recent comments:

1) "None of the possible reapers could have killed me" is not a premise! Pruss does not stipulate this. If you want a second premise (after your P1: "I am killed by a reaper"), you could say something like "only the first reaper can kill me, since any other would find me already dead". This is still unproblematic, and non-contradictory. It is only when we introduce the third premise -- "there are infinitely many Reapers in succession" -- that we get the contradiction, since that premise is incompatible with there being a first reaper.

Note that the issue is not that we have a hard time identifying the first one; it's that there literally cannot be one. It would be like saying we lack enough information to say what the smallest fraction is. There is no smallest fraction. Appeals to infinity or infinite decimal places are an aberration of post-Cantor mathematics; but, I can spot you all of that and you still can't get to the smallest fraction (i.e. the first Reaper).

2) Physical impossibility is an interesting thing to introduce. If you're saying there's some physical impossibility in infinitely many Grim Reapers (which aren't, to my knowledge, material entities), you'll have to spell it out. If it appeals to a problem with infinity, then... I mean, that's the whole point of this argument, isn't it? It's like when you said there might only be finitely many intervals between 8am and 9am, and then said that that would be begging the question.... Wouldn't it just be embracing the solution that Pruss gives (namely, Finitism)?

I confess that I have no idea what you're asking when you talk about the number line and "physically unrealizable circumstances". Do you mean that at tiny enough time intervals the scenario ceases to make sense? Just redo it with some instantaneous effect (like wave-function collapse or something) and massless particles. The "Grim Reaper killing Bob" stuff is just meant to make it humorous and memorable, right?

But, sure, if the scenario is physically unrealizable, that would also resolve things. It could still be the case that there would be a paradox if the scenario (or a similar enough one) could be realized physically. And wouldn't that counterfactual admission still justify embracing Finitism (in the absence of some other objection)?

Alexander R Pruss said...

"A paradox has two independent valid arguments leading to contradictory conclusions - take, for example, the Ross-Littlewood paradox, where the argument for the vase ending up empty does not entail that it ends up containing infinitely many balls, and vice-versa. We resolve a paradox by showing that one or the other horn does not hold (or possibly neither)"

Here is one argument:

1. You can have infinite backwards causal sequences. If you can have infinite causal sequences, you can have a Grim Reaper setup.

Here is a second:

2. You cannot have a Grim Reaper setup (for the reasons discussed).

I resolve the paradox by rejecting 1, and specifically by rejecting infinite backwards causal sequences.

ARaybould said...

Michael Gonzalez: Thank you for your reply - I think we are making some progress here.

Let me also add that I am not as certain about anything as I probably seem to be: I flip-flop almost hourly on whether there is really a paradox here and if so, what it is, but I tend to only post at those times when I feel that I have solved it to my satisfaction!

I take your point in your second paragraph - in fact, I think this version does a rather better job than convergent-series versions, as the latter seem vulnerable to the claim that it is mathematically respectable to say that the time of the killing is given by the limit of the series (in fact, I wonder whether that is why Alexander developed this version.)

With regard to "none of the possible reapers could have killed me": I was surprised to learn that I said it was anyone's premise, and on reviewing my prior posts, I see that I did not. In fact, I took it from your post, where it certainly seems to be presented as one side of what you see (or saw) as paradoxical: " if we deny [finitism], then it is both the case that I am killed by a Reaper before 9am and that none of the possible Reapers could have done it." What I did in my post was discuss the argument for that horn of the paradox.

I agree that "the issue is not that we have a hard time identifying the first one"; it is that we have no way to uniquely pick out the killer, even though we know there was one (and only one, if we deal suitably with the complication I mentioned in a footnote, that this formulation leaves open the possibility of more than one reaper waking at the same time. Let's say they draw lots in that case.) This is, as I said, an intended consequence of using an open interval: if it were simply changed to a (half)-open interval (which is only infinitesimally wider) I don't think anyone would find a paradox in it, as we could pick out the specific assassin, and the paradox seems to arise from people thinking there is an insurmountable problem if we cannot pick out the first one and say "this is the one who will do it."

"Appeals to infinity or infinite decimal places are an aberration of post-Cantor mathematics...." If you can demonstrate how we can uniquely identify each one of an actual infinity of reapers with a finite number of bits, I will reconsider the point you are referring to (information theory is math, even in the post-Cantor world.)
 
continued...

ARaybould said...

On physical realizability: you write "If you're saying there's some physical impossibility in infinitely many Grim Reapers (which aren't, to my knowledge, material entities)..." Ah, so they are not material? Can they kill without expending energy? Can their alarm clocks tick off time without expanding energy? Are reapers exempt from the relativistic speed limit? Can a corpus comprising of an actual infinity of reapers be assembled in finite time (such as since the Big Bang)? Is an actual infinity of anything physically realizable?  A rest-massless reaper might be just conceivable, but if it has any energy it must be travelling at light speed, and if it is, it has mass and time does not pass for it... It seems to me that you are taking the position that if we put aside all the reasons why the scenario is not physically realizable, then it is physically realizable!

I don't think your suggestion that we should "Just redo it with some instantaneous effect (like wave-function collapse or something)" will work out as you hope - so let's add quantum indeterminism to the mix! Then, possibly, we are not killed at all - now where is your paradox? Actually, if we are talking about physically-realizable Grim Reaper scenarios, we absolutely must include quantum indeterminism, because the physical world really is a quantum world - and if reapers are bosonic, maybe they did all do it!

Finally, I think the answer to the question in your last paragraph is no, at least if we are talking about time, rather than some abstract metaphysical concept that might have some resemblance to it.

ARaybould said...

Alexander, thank you for taking the time to read what we hoi polloi are writing at one another, let alone deigning to give us some pointers. I can imagine your eyes rolling as you look at the knots we tie ourselves up in!

I must say, however, that you have left me in your dust with this post. Firstly, given the quote you start with, I tentatively figure that your two numbered arguments are to be taken as the two sides of the paradoxical contradiction that Grim Reaper paradoxes set up.

Secondly, I can't imagine any way in which the execution (pun intended) of a Grim Reaper paradox involves backwards causal reasoning, whether infinite or finite. That leaves me guessing that you may be referring to the setup for a convergent-series Grim Reaper scenario, which seems to require infinite backwards-in-time reasoning to set the alarms. As far as I can tell, however, your version of the Grim Reaper scenario avoids this infinite-backwards-reasoning in its setup, which would (if my other assumptions about what you are saying here are correct) seem to save that form of Grim Reaper paradox from the no-infinite-reverse-causal-reasoning objection.

---

That aside, my response-du-jour to Grim Reaper paradoxes is that the paradox is purely mathematical, and has nothing to do with time. It goes like this:

Firstly, so long as we avoid attempting to identify the assassin by its time of awakening, I think we can come up with a consistent story: you are killed by the first reaper to wake, acting alone. We can still do that, and also identify the assassin, if the wakeup times are uniformly distributed over a closed interval - the killer is the one who wakes on the lower bound. We can arguably also do it for all convergent-series Grim Reaper paradoxes, if we accept the limit of the series as the time the assassin wakes (which is a mathematically-respectable choice, I believe.)

The problem of saying when the lone assassin woke comes with open intervals, because they don't have a least number. This does not have anything to do with time specifically, however, as we can have the same paradox without introducing time - for example, the sequence of rational numbers in the open interval (8,9) exists at 8.5 but not at 8, yet there is no end to be found anywhere, let alone between these numbers: we have a totally-ordered set with no end in either direction, yet it does not contain every number. Grim Reapers do not extend this paradox to time; they demonstrate it with a (hypothetical) example using time.

If there is a circumstance in which this mathematical paradox would have an observable consequence in the physical world, I would agree that it is a paradox in that aspect of the physical world in which it is manifest, but Grim Reaper paradoxes do not achieve that for time, or any other physical observable.

ARaybould said...

My apologies for posting so much, but I woke up this morning with what feels like a minor epiphany: in any Grim Reaper scenario, the act of the assassin 'converts' the open interval into a (half-)closed one, by identifying / defining / picking out the first reaper.

ARaybould said...

Actually, it is the setting of the alarm clocks that closes the interval. As soon as you have just one alarm clock set, you have an interval closed by a first time (which may be modified by the setting of another clock to an earlier time.)