an endless and hopeless quest for a pair of adjacent real numbers — keystone

Hold on a second! What is a "pair of adjacent real numbers"? There aren't adjacent pairs of rational numbers either. For any given pair of two distinct real (or rational) numbers, there are infinitely many more in between them.

Consider also that most statements of the paradox have the prisoner ignorant of the day on which they will be hung. The days are discrete and hence don't span a continuum. Furthermore, there is a finite amount of them: typically, the seven days of a particular week.

I think you're wrong to call it a sequel to the hanging paradox. It's conceptually nothing like the hanging paradox, other than the fact that both word problems both involve a hanging - which is just a superficial similarity. Conceptually it has more in common with zenos paradox of motion, which also has to do with problems of infinites and infinitely divisible units.

Well, I don't think the "paradox" even makes sense here. If he manages to find a moment when the guillotine is going through his neck (different example), the light should shine green. And the two alternatives have really nothing to do with the situation. Besides, it is not a paradox.

Hold on a second! What is a "pair of adjacent real numbers"? — Pierre-Normand

That's why I called it a hopeless task.

Consider also that most statements of the paradox have the prisoner ignorant of the day on which they will be hung. The days are discrete and hence don't span a continuum. — Pierre-Normand

Yeah, this is a fair point. But then again, I'm presenting it as a sequel, not a remake of the original.

I think you're wrong to call it a sequel to the hanging paradox. It's conceptually nothing like the hanging paradox, other than the fact that both word problems both involve a hanging - which is just a superficial similarity. Conceptually it has more in common with zenos paradox of motion, which also has to do with problems of infinites and infinitely divisible units. — flannel jesus

While I concur that this sequel shares conceptual similarities with Zeno's paradox, I diverge in the assessment that its resemblance to the hanging paradox is merely superficial. My contention is that unless the hanging paradox is approached from an eternalist perspective, a satisfactory resolution remains elusive.

In this sequel, the essence of surprise persists because nestled between any two real numbers exists another number—indeed, an infinite multitude of them. This signifies that amidst any observed moments lies a realm of unobserved moments wherein change could have transpired.

Dissimilar to the sequel's continuum of infinite points awaiting measurement, the original paradox comprises only five discrete points—Monday through Friday. From my eternalist standpoint, not all five of these points can be directly observed. While we ascertain his vitality on Sunday and his demise the ensuing Saturday, I posit that a fundamental limitation exists in our ability to fully witness all five points in the recording. An inherent barrier prohibits us from directly observing every point between his state of vitality and his demise, thus preserving the element of surprise.

My central contention is that change transpires when our gaze is averted. Indeed, such is the essence of reality's behavior (https://en.wikipedia.org/wiki/Quantum_Zeno_effect).

Well, I don't think the "paradox" even makes sense here. If he manages to find a moment when the guillotine is going through his neck (different example), the light should shine green. — Lionino

Yes, but he will never be certain that that moment is the first moment the guillotine is going through his neck.

Besides, it is not a paradox. — Lionino

I agree that there's nothing paradoxical about the sequel, but that's my point. I'm trying to present a perspective (an eternalist view) from which the original paradox is not paradoxical. I explained my view further in my last message to flannel jesus.

That's why I called it a hopeless task. — keystone

It gives your paradox a much easier angle of angle of attack than the unexpected hanging has. You can point out the error in the formulation. That "an adjacent pair of real numbers" doesn't exist.

I also wouldn't like to call it a sequel to the unexpected hanging because your riddle isn't (at least at face value) anything to do with induction (the mathematical reasoning concept) or self reference.

God then proposes a challenge: "If you're so convinced about the unfairness surrounding your demise, prove to me that you know the precise moment the noose first made contact with your neck. To aid you, I'll illuminate this light green for any instant you enter where the noose is in contact with your neck, or red if not." — keystone

This phrasing of the problem is effectively "a person will be guessing a finite or countable list of real numbers, they need to guess the exact start and/or end point of an interval of real numbers, why can't they?"

And that's got a few answers everyone would agree to. Countable sets ( the list of guesses ) are measure 0 in the reals, and moreover as @Pierre-Normand highlighted the reals are "dense" - there aren't adjacent numbers. So you can't detect the exact points a step change in a function ([0,1]->red,green) would occur by putting in any countable list - at least your guesses have probability 0 of being right if God's not been too human about it.

In that respect the kind of ambiguities in the framing are already solved problems, and it's a riddle about something completely different from the unexpected hanging.

It gives your paradox a much easier angle of angle of attack than the unexpected hanging has. You can point out the error in the formulation. That "an adjacent pair of real numbers" doesn't exist. — fdrake

Agreed, however, this is precisely why I find the sequel valuable. As I stated in my latest post to @flannel jesus, the sequel is not paradoxical because it's obvious that unobserved moments exist between any two instances, providing ample opportunity for surprise. Isn't it plausible that in the original paradox, a comparable scenario arises—between the states of being alive and deceased, there must exist unobserved moments. I understand this challenges our classical physics intuitions, but does it contradict our intuitions derived from quantum physics?

I also wouldn't like to call it a sequel to the unexpected hanging because your riddle isn't (at least at face value) anything to do with induction (the mathematical reasoning concept) or self reference. — fdrake

In this paradox, the details of the prisoner's punishment don't explicitly involve induction or self-reference. In the sequel, I offer a perspective in which the conditions of the initial sentence remain entirely logical. Through this lens, both induction and self-reference become redundant. These elements only emerge through the prisoner's flawed reasoning, rooted in classical physics. Therefore, it isn't the sentence itself that's paradoxical, but rather the prisoner's interpretation of it.

So you can't detect the exact points a step change in a function ([0,1]->red,green) would occur by putting in any countable list - at least your guesses have probability 0 of being right if God's not been too human about it. — fdrake

I believe there's a misunderstanding of my point. It's not enough for the prisoner to identify the initial moment when the light turns green. They must also pinpoint the final moment it was red. This step is crucial to confirm that the moment they observed was truly the first instance of the light turning green. Therefore, it's not about the prisoner's probability of finding a solution being zero; it's that a solution does not exist at all.

In that respect the kind of ambiguities in the framing are already solved problems, and it's a riddle about something completely different from the unexpected hanging. — fdrake

The main goal of the sequel is to offer a resolution to the original paradox, so I find myself at odds with that assertion. Essentially, the sequel's lack of paradox stems from the inability to witness every moment. My argument is that the initial paradox, much like the sequel, can be addressed by accepting the impossibility of observing every day. If one reviews the records and notes that the prisoner was alive on Monday but deceased by Wednesday, then choosing not to examine Tuesday's footage maintains the ambiguity of the exact time of death. On the other hand, if one decides to view Tuesday's footage and discovers the prisoner was already deceased, the only way to sidestep a paradox would be to alter the historical record, thereby erasing the observation from Monday. This means, in the revised history, the prisoner's death could have occurred on the now unwatched Monday or the scrutinized Tuesday. While this notion may seem far-fetched, it mirrors occurrences in the quantum realm: https://en.m.wikipedia.org/wiki/Delayed-choice_quantum_eraser.