Are Actual Loops of Objects Impossible?

Horace, Irene, and Julia are discussing what kinds of numbers and arrangements of objects are actually possible.

Horace comes up with a thought experiment with the following premise:

1. The Doomed Man is alive and surrounded by a loop of Grim Reapers. It is the Grim Reapers’ job to kill any Doomed Man — and they enjoy doing it!
2. Whenever a Grim Reaper sees the Doomed Man alive, it says to the one on its left, as a manner of courtesy, “After you,” deferring the task and closing its eyes. If there is no Grim Reaper on its left to which to defer, it does the deed itself.
3. Whenever a Grim Reaper is told, “After you,” it says to the one on its left, as a manner of courtesy, “After you,” deferring the task and closing its eyes. If there is no Grim Reaper on its left to which to defer, it does the deed itself.
4. Soon enough, the Doomed Man has been slain by one or more Grim Reapers.

It’s clear to all three friends that this situation is logically impossible. This is because even though each premise of the setup is conceivable individually, their combination is something both non-completable (via the endless deferrals from #3) and yet somehow complete per #4.

Horace concludes that the problem is found in #1: They are arranged in an actual loop.

The most elegant solution, he argues, is to instead arrange them in a queue, with a firm start and end. Once #1 is altered in this way, then each other conceivable premise can occur, and that’s nice, because it bothers us when we’re told a perfectly conceivable thing cannot occur for what seem like ephemeral reasons.

His conclusion: Actual loops of objects are impossible. (You may feel this is a strange takeaway.)

Irene, however, points out that we’re still in a pickle. That’s because an alternative #3 is perfectly conceivable: That the Grim Reapers have a manner that, if they lack a Grim Reaper on their left, they will instead defer to a Grim Reaper on their right (if there is one). Instead of an endlessly cyclic deferral, you get an endlessly zig-zagging deferral.

Her conclusion: Actual loops of objects are impossible and actual queues of objects are impossible, too. These must be theoretical only. (You may feel this is also a strange takeaway.)

Julia shakes her head.

Actual loops of objects are fine, Julia says. So are actual queues. Even actual infinities of objects are okay! The problem, Julia explains, arises when the Grim Reapers are infinitely deferential and yet #4 states the Doomed Man has been slain by one or more.

Now this doesn’t mean, clarifies Julia, that Grim Reapers cannot defer at all. If you have infinite Grim Reapers in a sequence who, while deferential, upon being deferred-to always do the deed themselves (vs. deferring a second time), then after a brief moment of courtesy the Doomed Man shall be slain.

Julia’s right. The root problem is not the number or arrangement of actual things after all.

Rather, the root problem is any scenario that stipulates that an endless job has come to an end.

Any locally-finite job is completable — at least, completable for magical beings we imagine — as long as that means genuinely locally-finite, lacking any dependencies upon the total number of objects sneaking into the job’s parameters.

You can have an actually endless racetrack in the real world. Just take the cross-cutting lines — the “ends” — off of a normal running track. But it’s not fair to ask someone to complete a race upon it, now robbed of its finish line. Which is to say that while the phrase, “The Running Man has finished the race,” is conceivable in isolation, its pairing with our end-deprived track is not.

Horace is predictably eager, upon hearing that, to blame the circularity of our track. He says to snap and stretch it into a straight course.

Irene is still concerned; what if the rules of the race are such that when you reach the ends, you have to turn around and go back the other way?

Julia sighs, and we know why: Whether the track is circular, straight and ended, or infinitely long, you can do a 100-meter dash. It’s all about the task “on top” of the actual objects, and whether or not you’ve defined it to be impossible.

More Info

In the below video from Joseph Schmid’s “Majesty of Reason” channel, Dr. Alex Malpass discusses this kind of “unsatisfiable pairing,” an observation which elegantly solves a number of stumpers, from the Grim Reaper Paradox to Thomson’s Lamp Paradox to Benardete’s Paradox of the Gods. While different folks have different takeaways from these and similar paradoxes, some use them to suggest that actual infinities are impossible (which in turn insinuates a First Cause) much as Horace used his bespoke Circle of Grim Reapers paradox to assert that actual loops of objects are impossible.