## Infinite Staircase Paradox

• 932
Given that in C3 X cannot be defined as either "0" or "1" but must be defined as either "0" or "1" then P3 is necessarily false. The supertask described in P3 is impossible.
That's clear as crystal. Your conclusion coincides with mine, so I'm perfectly happy with the argument.

P1. At t0 X = 0
C1. Therefore, at t1 X = 0
This puzzles me. Is this t(1) the same t as the t(1) in C3? It can't be. There must be a typo there somewhere.

One question, then - The state of X at any t(n), depends on its predecessor state at t(n-1), doesn't it? Isn't that a definition? Why is it inapplicable to t(1)?

I meant, that they can mislead us when we apply the principles to the activities of the physical world.
I think that's perfect. It's the conjunction of mathematics and - what can I say? - the everyday world.
What's difficult is the decision which is to give way - mathematics or the everyday world. Zeno was perfectly clear, but some people seem to disagree with him.

What is evident, is that we do not know how things move, and the exact "path" through space, that things take, whether they are big planets, stars and galaxies, small fundamental particles, or anything in between.
That suggests that we do know roughly how things move. I don't think that's what at stake in Zeno's thinking. His conclusion was that all motion is illusory. The only alternative for him was stasis. But I guess we can do better now.
• 14.4k
This puzzles me. Is this t(1) the same t as the t(1) in C3? It can't be. There must be a typo there somewhere.

No, it was three separate situations. Sorry if that wasn’t clear.

One question, then - The state of X at any t(n), depends on its predecessor state at t(n-1), doesn't it? Isn't that a definition? Why is it inapplicable to t(1)?

It is applicable to t1, but given the supertask described in P3 there’s no coherent answer to the definition of X at t1 (no final redefinition before t1) proving P3 to be impossible.
• 932
No, it was three separate situations. Sorry if that wasn’t clear.
Oh, I see now. You did explain, but I didn't pay enough attention.
Though I don't quite see how your B2 follows from your B1. But I don't think it is important.

It is applicable to t1, but given the supertask described in P3 there’s no coherent answer to the definition of X at t1 (no final redefinition before t1) proving P3 to be impossible.
You mean that we don't know the state of X at the last step before t(1), even though X must have been in one state or the other? (We don't have to work laboriously through each step. We just have to know how many there are steps there are between t(1/2) and the last step - we could work it out from that.)

It seems to me that we can work out the value of X for each and every step between t(0) and 1 if we work forward from t(0) but not if we try to work backward from t(1). In other words, whether X has a value at any stage depends on whether we define that stage in relation to the beginning or the end of the series. That seems very odd to me. But perhaps I've misunderstood. But I would be inclined to call a definition like that somewhat ill-formed.
• 14.4k
Though I don't quite see how your B2 follows from your B1.

It was redefined as 1 at t1/2 and never changed again, so is still defined as 1 at t1.

You mean that we don't know the state of X at the last step before t(1), even though X must have been in one state or the other?

There is no last step before t1, hence no coherent definition of X at t1. But also at no point between t0 and t1 is there a step where X goes from being defined (as either "0" or "1") to being undefined, and the definition of X is always retained until redefined to something else. It's a simple contradiction.

If you're trying to find a "solution" you won't find one. We just have to accept that supertasks are illogical. It's that easy.
• 932
If you're trying to find a "solution" you won't find one.
I'm not trying to find a solution, just to understand what's going on. Not so much why it's wrong, but why anyone would think it was right. Where does the illusion come from?

Given that in C2 X cannot be defined as either "0" or "1" but must be defined as either "0" or "1" then C1 is necessarily false. The supertask described in C1 is impossible.
I think I've just understood the significance of your A and B propositions. They are what justifies your formulation of the problem as a contradiction.

There is no last step before t1, hence no coherent definition of X at t1. But also at no point between t0 and t1 is there a step where X goes from being defined (as either "0" or "1") to being undefined, and the definition of X is always retained until redefined to something else. It's a simple contradiction.
If there is no last step before t1, there is no last-but-one step before the last step and no last-but-two step before that. And so on. The entire sequence unravels.
If you look at the series one way, it looks perfectly in order. If you look at it another way, it collapses entirely - it's not just a problem about defining the state of X at t1, but about defining the entire sequence.

A1. At t0 X = 0
A2. Therefore, at t1 X = 0

B1. At t0 X = 0 and then at t1/2 X = 1
B2. Therefore, at t1 X = 1

C1. At t0 X = 0 and then at t1/2 X = 1 and then at t3/4 X = 0, and so on ad infinitum
C2. Therefore, at t1 X = ?

Going back to your propositions A, B, C, it seems a fair guess that the problem is the insertion of "ad infinitum". That's the difference that causes X to become undefined. Our instinct that it should work derives from the fact that the series works perfectly well even if we do not insert any definite number of steps:-

D1. At t0 X = 0 and then at t1/2 X = 1 and then at t3/4 X = 0, and so on for n further steps where n is an even number.
D2. Therefore, at t1 X = 0

E1. At t0 X = 0 and then at t1/2 X = 1 and then at t3/4 X = 0, and so on for n further steps where n is an odd number.
E2. Therefore, at t1 X = 1

I think that's more or less what I was looking for.
• 1.4k
Yes, it affects how we think of them. It doesn't effect the situation, despite all the assertions to the contrary by several members.
— noAxioms
Yes - unless it is a fictional situation - whether in the philosophical or the literary sense.
I must disagree there. If there are two different descriptions of a fictional situation, and the description affects the thing being described differently, then they're describing two different things, not the same thing in two differnt ways.
The tortoise being overtaken is fiction, but mirrors real physical situations, unlike almost all the other examples in this topic. Describing the motion of Achilles as normal or as a supertask has zero effect on the ability of Achilles to overtake the tortoise.

A thought experiment is a valid method of deriving conclusions from premises. They only get fictional if the premises are faulty, such as the lamp, a device which cannot physically operate as described.
— noAxioms
That may explain why I have been confusing them. Thanks for that.
I must clarify that the lamp itself is physically impossible, making it fiction. I said 'faulty', which it is not. It measures something undefined, so it isn't a contradiction (a fault) that the final state isn't defined.

I have wondered whether one could replace the Thompson lamp with a question, such as whether the final number was odd or even.
Exact same scenario. But it's like asking if the smell of lavender is odd or even. There isn't a number that corresponds to the quantity of steps taken.

The lack of a first step does not prevent the beginning of the task
— noAxioms
It literally does.
This is exactly why I asked for your definition of 'start' since you seemed to be committing an equivocation fallacy between two definitions. You copped out and gave a synonym (begin) that has the same two definitions.
Is it Sv1 or Sv2? Because you are using both here, playing a language game.

I am saying that the lack of a first step does not prevent the beginning (Sv2 definition, 'transitions from not doing the task to doing it') of the task. You reply that it literally does, but Sv1 is the literal definition (the finite, 'has a first step').
I would not state that the lack of a first step does not prevent the first step from occurring (Sv1). That would be a contradiction indeed.

You ignored it and just said "when the time comes I say the next number". That doesn't explain how the recitation can begin without a first number to say.
You cannot show how that description doesn't work. Your only argument is that it doesn't perform a first step, but the description doesn't mention the need to do so, so the criticism is inapplicable.

I am right now trying to recite the natural numbers in descending order but am silent because I cannot begin.
You are not. It isn't a physically possible task. If you want to do a physically possible one, do Zeno's dichotomy. It's easy. You do it every day. The task is started despite the lack of a first step.

If you take the physical impossibliity away, then you failed because you didn't recite each number at the prescribed time. Your choice not to do so. You didn't follow my procedure. If it's been a minute, you admit to not following it.

Consider the infinite sequence {0, 1, 0, 1, 0, 1, ...}.

Now consider reciting its terms in reverse.
Undefined. You give no indication of when each number is to be recited. When do I say the 71st to the last zero for instance? I can answer that with a scenario that is properly described. It isn't a supertask as described.

Here's what Aristotle reported:

The second is the so-called 'Achilles', and it amounts to this, that in a race the quickest runner can never overtake the slowest, since the pursuer must first reach the point whence the pursued started, so that the slower must always hold a lead.
— Aristotle Physics 239b 14-17
That's apparently what somebody else reported about what Aristotle reported. I've seen it conveyed about 20 different ways. This particular wording says 'never' and 'always', temporal terms implying that even when more than a minute has passed, (we're assuming a minute here), Achilles will still lag the tortoise. The logic as worded here is invalid for that reason since the argument doesn't demonstrate any such thing. I've seen more valid ways of wording it (from Aristotle himself), in which case it simply becomes unsound.

How is this different from what I said?
It isn't much. I just didn't like the fact that the quote didn't match the site linked. Too bad Zeno's original argument is gone. Maybe he covered his ass better than the summary provided by somebody paraphrasing Mr. A.
• 12.5k
I think that's perfect. It's the conjunction of mathematics and - what can I say? - the everyday world.
What's difficult is the decision which is to give way - mathematics or the everyday world. Zeno was perfectly clear, but some people seem to disagree with him.

The difficult thing is that many human beings are like naive realists, and they think that our sense perceptions of "the everyday world" are a direct copy of the way an independent world would be. From this perspective, we cannot look toward the everyday world to be what needs to give way. But from a more philosophical perspective, we know that sense perception doesn't really show us the way the world is.

So to be prudent, I'd say both sides need to allow give and take. This may be like the ancient division between Parmenides with being and not being, and Heraclitus with becoming. Plato described how the two seemed to be fundamentally incompatible, and Aristotle provided principles whereby they both could coexist as different aspects of reality.

That suggests that we do know roughly how things move. I don't think that's what at stake in Zeno's thinking. His conclusion was that all motion is illusory. The only alternative for him was stasis. But I guess we can do better now.

That was Zeno's conclusion, from his paradoxes, that motion is impossible. But I do not think that this was what he was sincerely trying to prove. Clearly he could observe motion, and he would know that this would be considered a ridiculous proof. So I think his arguments were designed to show that there is incompatibility between motion as observed, and motion according to the principles of logic applied to it. Zeno came from the Eleatic school, so the first principle was "being", stasis, but what he was demonstrating was that this principle was insufficient to understand reality. That's why Socrates and Plato took interest in the sophistry of the Eleatics. The Eleatics could employ logic to prove absurd things, and this showed the gap between the "becoming" of the physical world, and the "being" of the Eleatics and Pythagorean idealism. So I think that Zeno, even though he came from the Eleatic school, was apprehending the faults in that ontology, and was sort of poking fun at it.

That's apparently what somebody else reported about what Aristotle reported. I've seen it conveyed about 20 different ways.

I quoted that directly from Aristotle's Physics. I gave the page and lines, 239b, 14-17. Further, Aristotle compares it to the arrow paradox, and says "the 'Achilles' goes further in that it affirms that even the quickest runner in legendary tradition must fail in his pursuit of the slower"

This particular wording says 'never' and 'always', temporal terms implying that even when more than a minute has passed, (we're assuming a minute here), Achilles will still lag the tortoise.

The time length is irrelevant. The pursuer will "always" lag the pursued, for the reasons indicated. The pursuer must reach the point where the pursued was, and in the time that it takes to do that, the pursued will move further ahead.

The logic as worded here is invalid for that reason since the argument doesn't demonstrate any such thing.

The logic is invalid for what reason? There is no specic time periods mentioned.

. I've seen more valid ways of wording it (from Aristotle himself), in which case it simply becomes unsound.

I gave you Aristotle's wording. He rejects the arrow argument which demonstrates that motion is impossible, by saying that time does not consist of instants. So that is an attack on the premises of that paradox. He then says that the solution to the 'Achilles' "must be the same". But he doesn't show how time not consisting of instants would solve the Achilles paradox. The matter of instants appears irrelevant here, and the problem seems to be with the assumed nature of space, rather than time.

I just didn't like the fact that the quote didn't match the site linked.

I can assure you, the quotes are taken directly from the referenced sites. I just went back to check. Click the links and you will see.
• 14.4k
It isn't a physically possible task.

It's not just physically impossible, it's logically impossible. No physics can allow me to begin reciting the natural numbers in reverse. I can't even say one number, let alone all of them. And this is true even if we're not reciting the natural numbers in reverse but the sequence {0, 1, 0, 1, ...} in reverse, i.e. where each term, individually, can be recited in less than a second.

That there is no first number to recite is the very reason that it is logically impossible to begin reciting them in reverse and it astonishes me that not only can't you accept this but you twist it around and claim that it not having a first number is the reason that it can begin without a first number.
• 14.4k
I'm not trying to find a solution, just to understand what's going on. Not so much why it's wrong, but why anyone would think it was right. Where does the illusion come from?

They're clearly being confused by maths. They think that because a geometric series of time intervals can have a finite sum and because this geometric series has the same cardinality as the natural numbers then it is possible to recite the natural numbers in finite time. Their conclusion is a non sequitur, and this is obvious when we consider the case of reciting the natural numbers (or any infinite sequence) in reverse.

There is a far more fundamental problem, and they're just ignoring it. I have no idea why. Perhaps because they can't look beyond the maths to what it would mean for us to actually carry out the tasks. This seems to be the mistake that Benacerraf made in his response to Thomson and which I addressed here.
• 932
They're clearly being confused (b)y maths.
.... and, as I think you must know, they think you are being wilfully dogmatic. That disagreement is what needs to be understood.
• 932
The difficult thing is that many human beings .... think that our sense perceptions of "the everyday world" are a direct copy of the way an independent world would be.
No, I don't think that they think that. It is a philosophical thesis. I'm not sure it is possible to articulate what people who have not thought about the question think the answer to it is.

But I do not think that this was what he was sincerely trying to prove.
So I think that Zeno, even though he came from the Eleatic school, was apprehending the faults in that ontology, and was sort of poking fun at it.
I don't think we have anything near the evidence required to divine Zeno's motives. We don't even have his articulation of the argument.

Clearly he could observe motion, and he would know that this would be considered a ridiculous proof.
Zeno came from the Eleatic school, so the first principle was "being", stasis, but what he was demonstrating was that this principle was insufficient to understand reality.
So I think that Zeno, even though he came from the Eleatic school, was apprehending the faults in that ontology, and was sort of poking fun at it.
But you don't know that he recognised what is so very clear to you, that the argument was ridiculous, or that he had "apprehended the faults in that ontology", though I admit that if he had understood what you understand, he might well have been poking fun at it. Still, other people since then have poked plenty of fun at it. But that's not a substitute for understanding the argument.

hat's why Socrates and Plato took interest in the sophistry of the Eleatics. The Eleatics could employ logic to prove absurd things, and this showed the gap between the "becoming" of the physical world, and the "being" of the Eleatics and Pythagorean idealism.

I agree that it is very likely that Plato/Socrates was addressing the apparent incompatibility of the perceived reality of change and the Eleatic rejection of that perception as illusory. The "two worlds" solution has its problems and, for my money, Aristotle's solution was much better.

The time length is irrelevant.
The exact length is indeed irrelevant. But the dimension of time is not. On the contrary, it is embedded in the argument.
• 5.9k
They're clearly being confused by maths. They think that because a geometric series of time intervals can have a finite sum and because this geometric series has the same cardinality as the natural numbers then it is possible to recite the natural numbers in finite time. Their conclusion is a non sequitur, and this is obvious when we consider the case of reciting the natural numbers (or any infinite sequence) in reverse.

I imagined you were arguing toward the claim that anything which has no first event is not physically possible. Which is a bit different from the geometric series angle. Which shows that there are logical possibles which have no first event. The sticking point between you and us series heads seems to me that you've been arguing on the basis of "no first event" blocking logical possibility, which would then block physical or metaphysical possibility. Which is why I intervened, because while I agree with your argument strategy, the means you were arguing for it were imprecise enough to admit the geometric series counterexample. I do think your points can be steelmanned though.

You might come up with a mathematical model for "recitation of first element", which maps a series to its minimum or maximum. Since the series has a minimum, you can do that. It has no maximum, so you can't. It's the same principle as asking someone to count backwards from infinity. And then it's true, it's not logically possible to "recite" backwards from infinity, or "recite" the maximum of an increasing geometric series.

Going back to the first page of the thread, such a "recitation" for the state of Thompson's lamp, or just isolating the "state", could be construed by taking a time period and associating it with the states the lamp takes in that time period in order. If Thompson's lamp has states in a time period, they'll be picked out by that. However, the function which generates the values of Thompson's lamp has the property that for every time period X, there exists a time period Y such that if max( Y )>max( X ) then Y contains at least two states (on or off). You get those by going further toward the completion time. That property implies there is simply no "state" of the lamp at limit of 2 minutes. So it having a state is logically impossible.

What makes Thompson's lamp a paradox, then, is a physical or metaphysical intuition about the concept of the state of the lamp. There needs to be a beginning to the process, and it needs a unique isolable end state. Both the geometric series and Thompson's lamp have no unique isolable end state.

That will then bottom out in an inquiry about whether there are physical or metaphysical possibles which have non-unique and non-isolable end states.
• 12.5k
That there is no first number to recite is the very reason that it is logically impossible to begin reciting them in reverse and it astonishes me that not only can't you accept this but you twist it around and claim that it not having a first number is the reason that it can begin without a first number.

NoAxioms has a habit of making astonishing claims, then instead of recognizing the incorrectness, arguing some twisted principles. Like above, noAxioms insisted Zeno did not conclude that the faster runner could not overtake the slower, then refused to recognize my references, insisting they were in some way improper.

They're clearly being confused by maths. They think that because a geometric series of time intervals can have a finite sum and because this geometric series has the same cardinality as the natural numbers then it is possible to recite the natural numbers in finite time. Their conclusion is a non sequitur, and this is obvious when we consider the case of reciting the natural numbers (or any infinite sequence) in reverse.

This is the problem with mathematical axioms in general. As fishfry said, I can't really count the natural numbers, but I can state an axiom that the natural numbers are countable, and this counts as me having counted the natural numbers. So mathematicians really need to be careful to distinguish between the fantasy world they create with their axioms, and the true nature of what is "logically possible". Just because it can be stated as an axiom does not mean that it is logically possible. And when the axiom claims that something which is by definition impossible (to count all the natural numbers), is possible, then there is contradiction, therefore incoherency, inherent within that axiom. But would a mathematician accept the reality of an axiom which is self-contradicting?

There is a far more fundamental problem, and they're just ignoring it.

The problem is the age-old incompatibility between being and becoming. Logic, and this includes mathematics, applies naturally to "what is", "being". But "becoming" has aspects which appear to escape logic, what lies between this and that, one and two, etc., and therefore it seems to be illogical. If we apply the logic of being, to the reality of becoming, we find paradoxes as Zeno demonstrated.

This implies that "becoming" requires a different form of logic. That's what Aristotle laid out with his definitions of "potential", and "matter", as the aspects of reality which violate the law of excluded middle. In modern times, much progress has been made with modal logic, and probabilities. But the truth is that these aspects of reality, those which are understood through probability, remain fundamentally illogical, and the so-called "knowledge" which is derived creates an illusion of understanding.

I'm not sure it is possible to articulate what people who have not thought about the question think the answer to it is.

It's simple, talk to people, ask them. Then you'll see that it's more than just a matter of thinking about the question, it is a matter of making the effort to educate oneself. Metaphysics is not apprehended as a valuable subject.

I don't think we have anything near the evidence required to divine Zeno's motives. We don't even have his articulation of the argument.

Well, there is a lot of information available from Plato. In works like "The Sophist" and "The Parmenides", he takes a very critical look at the motives of some of the Eleatics, Zeno in particular. Specifically, in "The Sophist" he attempts the very difficult task of developing a distinction between philosophy and sophistry, even the sophist is engaged in philosophy.

But you don't know that he recognised what is so very clear to you, that the argument was ridiculous, or that he had "apprehended the faults in that ontology", though I admit that if he had understood what you understand, he might well have been poking fun at it. Still, other people since then have poked plenty of fun at it. But that's not a substitute for understanding the argument.

It's very clear from the discussion at the time, Plato and Aristotle, that Zeno knew he was using logic to produce absurd conclusions. There should be no doubt in your mind about that. He did not pretend to believe what he had proven, that motion is impossible, that the faster runner could never overtake the slower, etc..
• 932
I must disagree there. If there are two different descriptions of a fictional situation, and the description affects the thing being described differently, then they're describing two different things, not the same thing in two different ways.
I see your point. But you must know that there is a great deal of philosophy around your view of this. But I won't try to drag you through it, is because I'm not sure how relevant it is. Yet.

The tortoise being overtaken is fiction, but mirrors real physical situations, unlike almost all the other examples in this topic. Describing the motion of Achilles as normal or as a supertask has zero effect on the ability of Achilles to overtake the tortoise.
I agree with that. So when someone describes the situation in a way that seems to make that fact impossible, why don't we just reject it as inapplicable?

I must clarify that the lamp itself is physically impossible, making it fiction. I said 'faulty', which it is not. It measures something undefined, so it isn't a contradiction (a fault) that the final state isn't defined.
But we allow physical impossibilities into fiction all the time. They even crop up in philosophical examples. "The sun might not rise tomorrow morning". "Twin Earth has water that is not H2O". I won't even mention philosophical zombies, brains in vats or simulations.
Your point about the final state not being defined is about logic, not physics (despite some people thinking that it is about physics).
In any case, the final state is defined. It must (on or off) or (0 or 1). Wouldn't it be more accurate to say that it is undetermined? Or is the final state the one immediately preceding the limit; in any case, it is not determined. So is the one before that.... But it would be absurd to say that every state in the series is indeterminate. It seems that whether anything here is determined is a question of how you look at it - from the beginning or from the end.
• 1.8k
Therefore I cannot start.

True. And that implies time is discrete how?

Going back to the first page of the thread, such a "recitation" for the state of Thompson's lamp, or just isolating the "state", could be construed by taking a time period and associating it with the states the lamp takes in that time period in order. If Thompson's lamp has states in a time period, they'll be picked out by that. However, the function which generates the values of Thompson's lamp has the property that for every time period X, there exists a time period Y such that max( Y )>max( X ) contains at least two states (on or off). You get those by going further toward the completion time. That property implies there is simply no "state" of the lamp at limit of 2 minutes. So it having a state is logically impossible.

What makes Thompson's lamp a paradox, then, is a physical or metaphysical intuition about the concept of the state of the lamp. There needs to be a beginning to the process, and it needs a unique isolable end state. Both the geometric series and Thompson's lamp have no unique isolable end state.

Good post
• 14.4k
True. And that implies time is discrete how?

If time is continuous then supertasks are logically possible. Supertasks are logically impossible. Therefore, time is discrete.
• 1.8k
If time is continuous then supertasks are logically possible.

Time being continuous is necessary but not sufficient for any given supertask being possible. Supertasks being impossible (especially the specific one you brought up) does not imply time is not continuous.

Any given example does not prove that supertasks in general are necessarily impossible. If the common necessity among all supertasks is time being continuous, the only way to prove the impossibility of supertasks is to prove time is not continuous.

(a and b and c and d) → supertask
not supertask does not imply not a
• 14.4k
Any given example does not prove that supertasks in general are necessarily impossible.

I addressed this here and here.
• 1.8k
Given that in C2 X cannot be defined as either "0" or "1" but must be defined as either "0" or "1" then C1 is necessarily false. The supertask described in C1 is impossible.

Me and fishfry have insisted that this is a case of missing limit. I saw your post here, the only reply to it is noAxiom's, to which then there was no reply, only a mention by fdrake. I haven't checked your post with the "refutation" of Benecerraf yet, neither do I think noAxiom's post addresses it fully.

When it comes to this post,
Given that in C2 X cannot be defined as either "0" or "1" but must be defined as either "0" or "1" then C1 is necessarily false. The supertask described in C1 is impossible.

Is the failure of C2 really the consequence of the impossibility of C1, or is there an unstated premise?

In all cases the definition of X at t1 must be a logical consequence of what occurs between t0 and t1.

What so many people disagree on is that. You think that the end of the sequence at t=1 is a temporal/logical consequence of what happens before. Others don't think so, because the lamp is being infinitely redefined until t=1.
• 14.4k
Me and fishfry have insisted that this is a case of missing limit.

That's why it's impossible to complete.

You think that the end of the sequence at t=1 is a temporal/logical consequence of what happens before.

Yes, I address that here.
• 1.8k
You are interested in exploiting that to define metaphysics. Perhaps that works, perhaps it doesn't

It was more of taking the phrase "metaphysically (im)possible" to mean "there is (not) a possible world where" and seeing where that leads. And if it leads anywhere is that maybe the definition of metaphysically possible is «that which follows the rules of the game». That seems abusive of the meaning of the words, or the words are not well-defined (many would say so for "metaphysics").
One of the first replies of the thread is this:
By Chalmers, logical = metaphysical; by Shoemaker, metaphysical = physical.
If jorndoe is representing the view well, I am confident both have good reasons to make such equations; I was exploring ways to make the semantics of "metaphysical" not fully overlap with "logical" or "physical".
• 1.8k
Yes, I address that here.

I understand the intuition you use to affirm that argument, I imagine others do too. At t=1 the sequence has ended, and the lamp must be either on or off. You use the same premise on it:
The status of the lamp at t1 must be a logical consequence of the status of the lamp at t0 and the button-pressing procedure that occurs between t0 and t1 because nothing else controls the behaviour of the lamp.
People disagree with the premise because we are not confident we can use such intuitions when the — unintuitive — concept of infinity is involved.
• 14.4k
People disagree with the premise because we are not confident we can use such intuitions when the — unintuitive — concept of infinity is involved.

The important part from that post is this:

The fallacy in his reasoning is that it does not acknowledge that for all tn >= t1/2 the lamp is on iff the lamp was off and I pressed the button to turn it on and the lamp is off iff the lamp was on and I pressed the button to turn it off.

If the lamp is on at t1 then it must have been either turned on at t1 or turned and left on before t1, neither of which are allowed given the supertask, hence the contradiction.
• 932
By Chalmers, logical = metaphysical; by Shoemaker, metaphysical = physical.
It would be a mistake to apply (((P = Q) & (Q = R)) implies (P = R)) without checking very carefully whether "Q" means the same for both of them. It is not something one could take for granted. I wouldn't take that thesis seriously without cross-questioning the author very carefully.

It was more of taking the phrase "metaphysically (im)possible" to mean "there is (not) a possible world where" and seeing where that leads. And if it leads anywhere is that maybe the definition of metaphysically possible is «that which follows the rules of the game». That seems abusive of the meaning of the words, or the words are not well-defined (many would say so for "metaphysics").
I doubt if it is possible to abuse the word "metaphysics". It has been abused so often in the past. In fact, it is so ill defined that I'm not sure what would count as abuse.
Three points:-
I have problems with the term "synthetic necessity" because I don't understand what that does to the meaning of "contingent". (I'm taking the Kripke-style interpretation that it means "In any world in which ...., this must be the case." - and in "in any world in which knock-out tournaments are played, it cannot be the case that two opponents in round 1 can meet each other again in round 2".) Tempting as it is, since logic is also a (language) game, or at least has rules, if metaphysics is that which follows the rules of the game", it aligns metaphysics with logic. But I do admire Toulmin's argument and recognize that he identifies an important class of propositions that have not figured much in philosophy.
I'm afraid I understand the possible worlds interpretation of possibility as simply possibility. Either way, of course, that aligns metaphysics with logic.
Many of the uses of apparently metaphysical language seem to me to be a simple matter of using what logic describes as "de dicto" as "de re" - possibly without being aware of what they are doing.
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It's simple, talk to people, ask them.
If I knew how to ask without leading them into philosophy, I would.

Well, there is a lot of information available from Plato.
The Stanford Encyclopedia is the best quick reference that I know of for something like this.

Almost everything that we know about Zeno of Elea is to be found in the opening pages of Plato’s Parmenides. There we learn that Zeno was nearly 40 years old when Socrates was a young man, say 20. Since Socrates was born in 469 BC we can estimate a birth date for Zeno around 490 BC. Beyond this, really all we know is that he was close to Parmenides (Plato reports the gossip that they had a sexual relationship when Zeno was young), and that he wrote a book of paradoxes defending Parmenides’ philosophy. Sadly this book has not survived, and what we know of his arguments is second-hand, principally through Aristotle and his commentators
SEP - Zeno's paradoxes
From what I could find, Aristotle has very little about Zeno and nothing about his motives. But what he does summarize (some of) the arguments, which Plato doesn't.
I hadn’t realized quite how close in time they were. It seems that the scenario in the Parmenides, which seems to be far and away the best source we have, could have taken place. Not that we know that it did. On the face of it Plato is not an implausible source – if only separating out the history in Plato was not so complicated.

The evidence surveyed here suggests that Zeno’s paradoxes were designed as provocative challenges to the common-sense view that our world is populated by numerous things that move from place to place.
No evidence of your interpretation here.

Thus, while Zeno accepts Socrates’ point that his own arguments aim to show that there are not many things, he corrects Socrates’ impression that, in arguing this point, he was just saying the same thing as Parmenides in a different form.
Or here.

Plato’s references thus consistently connect Zeno with the rise of eristic disputation, and it is perfectly plausible that his arguments against plurality and motion would have been well-known examples of making the weaker case seem the stronger.
Now, this is another example of what I was talking about. Plato (and others) were confident that Zeno’s case was weak. Fair enough, but to go on, as Plato does, to accuse the sophists of deliberate deception or wilful blindness is completely unjustified (except when, as in the Protagoras,(?) Gorgias (?) someone boasts about doing so – though it doesn’t follow that everyone that Plato accuses of rhetoric and sophistry did so boast.). I have seen it often before, particularly in the last year on these forums. But it is most disheartening.

Zeno’s influence, however, on the great sophists who were his contemporaries and, more generally, on the techniques of argumentation promulgated among the sophists seems undeniable.
But accepting that connection is a long way from accepting that he had any doubts about the validity of his conclusions.

Zeno was not a systematic Eleatic solemnly defending Parmenides against philosophical attack by a profound and interconnected set of reductive argumentations. Many men had mocked Parmenides: Zeno mocked the mockers. His logoi were designed to reveal the inanities and ineptitudes inherent in the ordinary belief in a plural world; he wanted to startle, to amaze, to disconcert. He did not have the serious metaphysical purpose of supporting an Eleatic monism” (Barnes 1982, 236).
I was wrong about that. I elided Parmenides with the Eleatics, though the difference is, perhaps, somewhat metaphysical (!). However, the difference matters when it comes to Zeno, so now I can get it right. It does not follow that Zeno did not believe that his conclusions were not true.

All the quotations above are from SEP - Zeno of Elea
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If jorndoe is representing the view well, I am confident both have good reasons to make such equations; I was exploring ways to make the semantics of "metaphysical" not fully overlap with "logical" or "physical".
Well, whatever prompted you, the project makes sense to me and I agree with Toulmin. I'm not convinced about the relationship of those propositions with metaphysics or their classification in the analytic/necessary/a priori constellation. However, preserving those concepts doesn't seem to me particularly important. I would be quite happy to abandon all of them.
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I accept this:

P1. If we can recite forward 1, 2, 3, ... at successively halved intervals of time then we can recite all natural numbers in finite time

But I reject these:

P2. We can recite forward 1, 2, 3, ... at successively halved intervals of time
C1. We can recite all natural numbers in finite time

Given P2, what is the first natural number not recited? I seem to remember having asked you this several times already.

If you want to claim that C1 is true then you must prove that P2 is true. You haven't done so.

What is the first number not recited?
• 2.9k

Any initial step necessarily leaps over all but finitely elements of the sequence. Same reason that any neighborhood of the limit of a sequence contains all but finitely many elements of a sequence.
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This is what I mean by reciting backwards:

If I recite the natural numbers <= 10 backwards then I recite 10, then 9, then 8, etc.
If I recite the natural numbers <= 100 backwards then I recite 100, then 99, then 98, etc.

If I recite all the natural numbers backwards then I recite ... ?

It's self-evidently impossible. There's no first (largest) natural number for me to start with.

I don't think you and I are making progress.

I have agreed repeatedly that we can't "count all the natural numbers backwards" since an infinite sequence has no last element.
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You are right that the historical contingency should make us suspicious. (Descartes, by the way, has a description of statues "animated" by a hidden hydraulic system - I think in Versailles). But I don't think the process is simply over-enthusiastic. It seems reasonable to try to apply a new discovery as widely as possible. That way, one discovers its limitations.

Yes ok, but that supports the possibility that in the future, our current preoccupation with "mind as computer" will look as dated as "mind as waterworks" of the Romans.

So the VR theory doesn't solve anything at all, it leaves the mystery of what my own consciousness is.
— fishfry
That's more or less one Ryle's favourite arguments against dualism.

That it explains nothing? I agree. Like saying "God did it." Or saying the Great Sky Computer (GSC) did it. Except that God is not restricted to being a computation, whereas the GSC is, making God a less unreasonable hypothesis.

The sequence 1/2, 1/4, 1/8, ... also has a limit, namely 0, and no last element. But if you put the elements of the sequence on the number line, they appear to "come from" 0 via a process that could never have gotten started. This is my interpretation of Michael's example of counting backwards.
— fishfry
Clearly "<divide by> 2" is not applicable at 0.

Well you never "reach" 0, but 0 is the limit.

Would it be right to say that "+1" begins at 0 and has no bound and no limit, and that "<divide by> 2" begins at 1 and has no bound, but does have a limit? But they both they have a defined start and no defined end.

If you allow the transfinite ordinals, the sequence 1, 2, 3, ... has the limit $\omega$. And even if this seems unfamiliar, it's structurally identical to the sequence 1/2, 3/4, 7/8, ... having the limit 1, which is much more familiar.

Without axioms it's difficult to get reasoning off the ground. You have to start somewhere, right?
— fishfry
Yes. The difficulty is how to evaluate a starting-point. True or false isn't always relevant. Which means that it can be difficult to decide between lines of reasoning that have different starting-points.

What is the starting point of no axioms? It's like playing chess with no rules.
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