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But which is not defined.

It's more than that; the lamp can't be on and can't be off, even though it must be one or the other. This is a contradiction, and so therefore the supertask is proven impossible in principle.
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It's more than that; the lamp can't be on and can't be off, even though it must be one or the other. This is a contradiction, and so therefore the supertask is proven impossible in principle.
Does the difference matter?
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But mixing up actual stairs with models of stairs just produces a confusion, so the paradox is just an illusion - in my opinion.

Exactly. There is no paradox caused by an infinite staircase, because an infinite staircase is a square circle, barely conceivable if conceivable at all.

What if every time I bit an apple and ate it, in the time I chewed the bite, the apple grew in size bigger than the bite I just took? Do I need to figure out the math here to see how to prevent the apple from eating me?

Despite the staircase being endless, he reached the bottom of it in just a minute.

Staircases are always, and only, actually, finite, as any object is. The endlessness of the staircase is brought to an end at the bottom, so it is not endless, so there is no sense to the word “despite”.

This is not a paradox, but a confusion of concepts (like the number 1 or infinitely) with actual things (like a one step down one stair, and never reaching the bottom or doing so in a minute).
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This is not a paradox, but a confusion of concepts (like the number 1 or infinitely) with actual things (like a one step down one stair, and never reaching the bottom or doing so in a minute).

:up:
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There are two cases in play at the moment - "0, 1, 0, 1, ..." and "1/2, 1/4, 1/8, 1/16.." Comments switch between them without always being clear. You are, however, quite right that the first sequence doesn't have a limit and the second one has what we could call a natural limit.

I know I'm right :-) Now if I can just get @Michael to agree!
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Now if I can just get Michael to agree!

I have always agreed that the sequence "0, 1, 0, 1, ..." does not converge.

I disagree with your claim that with respect to Thomson's lamp we can simply stipulate that the lamp is on after two minutes. See my previous post and my initial defence of Thomson on page 13.
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I disagree with your claim that with respect to Thomson's lamp we can simply stipulate that the lamp is on after two minutes.
@fishfry will speak for himself. But I think the point is that, even a convergent sequence, which does have a limit, does not have a end or last step defined - indeed, is defined as not having one. That means that any answer whatever is equally valid and invalid.
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That means that any answer whatever is equally valid

That's not true, as I explained here, and as I alluded to above. It is not just the case that whether the lamp is on or off after two minutes is undefined but that the lamp cannot be either on or off after two minutes.

As Thomson says in his paper, "the impossibility of a super-task does not depend at all on whether some vaguely-felt-to-be associated arithmetical sequence is convergent or divergent."
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That's not true, as I explained here, and as I alluded to above. It is not just the case that whether the lamp is on or off after two minutes is undefined but that the lamp cannot be either on or off after two minutes.
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It is not just the case that whether the lamp is on or off after two minutes

I don’t understand. How do you ever arrive at the two minute mark?
1 minute, half a minute later, quarter minute later than that, etc., infinitely…you never arrive at the two minute mark.

Like the endless bottomless staircase that for some reason had a bottom with a dead guy.

??? Two minutes and a bottom step subvert the issue and banish infinity from the math of it.

These are manufactured conundrums mixing what is an actual occurrence (walking down a step, turning a light on or off) with what is not an actual occurrence but a conceptual/mathematical idea that has no weight or influence on walking or flicking light switches.

There no paradox because the lamp can’t be on or off. There is just an endless motion (invented in a mind that can invent things and objects that can’t actually be built and tested).

If you are going to ponder whether the light is actually, physically on, you have to ponder whether a switch can be switched as rapidly as you would have to switch it as you approached two minutes. The light might be off because you broke the switch. I know, to a mathematical/logical student I am missing the point, but then, if we can assume functioning switches to lay out the mathematical intrigue, why can’t we assume the mathematical intrigue of never reaching two minutes and therefore never able to determine an on/off state at the two minute mark? And conclude that such a scenario builds a lamp that will endlessly turn on and off at ever increasing speeds with no limit or single state thereby enlightening or darkening said limit.

As you approached 2 minutes with really small fractions (but never got there if you are being consistent), would the speed between switching the light on and off eventually approach and then exceed the speed of light? (That’s one solid switch!). So how would we be able to tell whether the light we saw was the light from the latest switching motion or light from a few switches ago? We’d need a new measure besides our eyes, and could not watch the experiment in real time anymore.

We can’t mix real things with infinity as a limiter. Real things always include finite limits. The positing of infinite physical steps, or infinite half steps, pose logical perplexities, but not actual paradoxes.
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I don’t understand. How do you ever arrive at the two minute mark?

I don't know what you mean by "arrive" at the two minute mark. Two minutes just pass. That's how the world works.

Imagine I am facing a clock with my back to a lamp. The experiment starts at 12:00. Through some automated process the lamp turns on and off at successively halved intervals of time. When the clock shows 12:02 I turn around. Is the lamp on or off?
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That's how the world works.

Haha! Ok, so the switch clicks at 1 minute, then it clicks again at 30 seconds more, then it clicks again at 15 seconds more, then again at 7.5 seconds, etc.

What will the switch and light be doing at 2 minutes?

Nothing in relation to the switch as described above. Something beyond the parameters the initial scenario set in motion.

It may be on. It may be off. It will probably be broken or melted. But in relation to the switch described above those possible outcomes have no bearing because the switch was designed to serve a function before two minutes time can elapse.

If the light clicks on at two minutes, or off, either the switch malfunctioned, or it can do more then switch itself at half intervals of the prior lapsed time.

“When the clock shows 12:02 I turn around. Is the lamp on or off?”

Whichever it is, it wouldn’t be a function of a switch that operates by waiting half the time to switch and then half of that time to switch again, and then half of the half, etc, because such a switch would have no direction or programming to follow at two minutes, because it is designed to operate before two minutes could pass.
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The lamp is off. After one minute the lamp turns on. Is the lamp on or off after two minutes? It's on because it was turned on after one minute and then never turned off again.

If you can't apply the same reasoning to the lamp having turned on and off an infinite number of times before the end of two minutes then you must accept that it makes no sense for the lamp to have turned on and off an infinite number of times before the end of two minutes.
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The same is true if the lamp switches just the once

No, that’s a different equation. That switch is more easily predicted. The switch on Thompson’s lamp requires some serious calculation to determine its state after say one minute fifty-nine seconds. It can be calculated though. But it can’t be calculated at 2 minutes time, because it was designed to function within two minutes. Once two minutes lapse, the switch’s function has no relationship to whatever would be the state of affairs (which I’m telling you would likely be a fire, so in a sense, I’m guessing the lamp would be on).

So we resolve the paradox by accepting the metaphysical impossibility of supertasks.

I’d say we never had a paradox. There is no paradox at 2 minutes, because by the time we get to 2 minutes, the initial switch and algorithm it functioned by are no longer in play, unable to be held at odds with whatever might be the case at 2 minutes.
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The lamp is off. It turns on and off only as described:

Scenario 1
The lamp turns on after 1 minute.

Is the lamp on or off after 2 minutes? It's on.

Scenario 2
The lamp turns on after 1 minute and off after a further 30 seconds.

Is the lamp on or off after 2 minutes? It's off.

Scenario 3
The lamp turns on after 1 minute, off after a further 30 seconds, and so on ad infinitum.

Is the lamp on or off after 2 minutes?

In all three scenarios the switch is "designed to function within two minutes."
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Is the lamp on or off after 2 minutes?

In all three scenarios the switch is "designed to function within two minutes."

Not if you want to answer the question in scenario three. Or more precisely, not designed to function at or after two minutes.
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Or more precisely, not designed to function at or after two minutes.

That's also true about the first two scenarios – neither switches after 1 minute and 30 seconds – and yet we can still answer the question about the lamp after 2 minutes, and even after 2 years.
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So what are we disagreeing about?

Whether this is paradox, or whether there is an answer to question 3?

I don’t think there is an answer to question 3. Because the switch is not designed to ever present the question.
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Because the switch is not designed to ever present the question.

I don't know what you mean by this.

Given that the lamp must be either on or off after two minutes we must ask the question. If you cannot provide a coherent answer then you must accept that your premise – that the lamp has turned on and off an infinite number of times – is necessarily false, and so that supertasks are metaphysically impossible.
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Given that the lamp must be either on or off after two minutes

How is that? How is it on or off at or after two minutes?

It cannot be a function of a switch that operates by switching every half of the prior interval. Some other function needs to be introduced into the picture to ask about the state of the lamp at 2 minutes or beyond.

Why is it given by this switche’s function that the lamp will be anything in particular at or after 2 minutes? You are just assuming something exterior to the premise about time and lamps.

In that case, if the lamp must be on, or off, at two minutes, is that state caused by the switch?
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How is that? How is it on or off at or after two minutes?

Because it's a lamp. If it exists at 12:02 then it's either on or off, and it exists at 12:02.

It cannot be a function of a switch that operates by switching every half of the prior interval.

And that's precisely why supertasks are impossible.

We are imagining, for the sake of argument, that we are in some alternate universe with physical laws that allow us push a button to turn the lamp on and off at successively halved intervals of time. We want to know what happens to the lamp if we do this (and only this).

As you say, we don't get to "just [assume] something exterior to the premise about time and lamps."

So the only premises we are allowed to work with are:

P1. If the lamp is off and the button is pushed then the lamp is turned on
P2. If the lamp is on and the button is pushed then the lamp is turned off
P3. The lamp is off at t0
P4. The lamp is either on or off at t1
P5. The button is pushed at successively halved intervals of time between t0 and t1

So we ask: is the lamp on or off at t1?

If you cannot provide a coherent answer then you must accept that the premises are inconsistent. Given that there's nothing problematic about P1 - P4, we must accept that the problem lies with P5. It is necessarily false. Supertasks cannot be performed.
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And that's precisely why supertasks are impossible.

And that’s precisely why the question of whether the lamp will be on or off at two minutes will never present itself.

If you want to talk about how lamps and intervals of time work in an alternate universe why are we using any premises from this universe about lamps and so on?

The three stooges were hungry and looking for ideas on what to eat. Larry says, “if we had ham, we could have ham and eggs, if we had eggs.” Mo hits him on the head.

If we want to ponder what a lamp will be doing at two minutes time, we could just hook it up to a switch that switches every half of the interval of time prior, if we had a universe where such a switch existed.

This conversation (and not your fault) is starting to feel like a descent on a bottomless staircase.
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And that’s precisely why the question of whether the lamp will be on or off at two minutes will never present itself.

It does present itself because the lamp must be either on or off after two minutes as per the law of excluded middle.

We want to know what would happen to the lamp if we were to push the button to turn it on and off at successively halved intervals of time within two minutes. Nothing else happens to the lamp except what we cause to happen to it by pushing the button.

Is it left on or off after we stop pushing the button? If you can’t answer this then you must accept that supertasks are metaphysically impossible.
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at successively halved intervals of time within two minutes

Then stop talking about at two minutes or after two minutes. That’s some other scenario.

Don’t you see that?

Two minutes or more is outside the universe of “at successively halved intervals of time within two minutes.”

The question “is the lamp on or off at or after two minutes” is talking about some other lamp.

Maybe you are really picturing a person pushing a button? Do you think you could push a button to work the premise of switching the lamp at successive half intervals? A lamp switch like this would have to be automated, and it would not need any instructions or programming at or beyond two minutes to function.

I feel like I just suggested ham and eggs again.
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the lamp cannot be either on or off after two minutes.

That's what I've been arguing since the beginning of the thread. And the reason why it can't be on or off is that in the described 'possible world' two minutes cannot pass. So the possible world in which "after two minutes" makes sense is incompatible with the possible world of that lamp.

I don’t understand. How do you ever arrive at the two minute mark?
1 minute, half a minute later, quarter minute later than that, etc., infinitely…you never arrive at the two minute mark.

You are exactly correct. In the described scenario, the stipulated possible world, the two minute mark cannot be reached. It's a Zeno-type argument. Michael insists that two minutes will pass, but that's a reference to a different, incompatible, possible world, the one designed around our empirical experience.

Two minutes just pass. That's how the world works.

See, you are referring to a different possible world here, the one derived from empirical experience. But this possible world is inconsistent with the one that the lamp is in.

Then stop talking about at two minutes or after two minutes. That’s some other scenario.

Don’t you see that?

Believe me Fire Ologist, I tried my best. Two months later Michael is still stuck in the same infinite loop.
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Two months later Michael is still stuck in the same infinite loop.

I posted a bit earlier in in the thread, then noticed recently it was up to 23 plus pages so I was curious if everyone had gotten to the bottom of the staircase.

But found out we are still counting the same steps over and over again!

I think I’ll give it another two minutes then I’m done.

Possibly done for an infinite number of minutes.

At that point I might post again…apparently.
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I have always agreed that the sequence "0, 1, 0, 1, ..." does not converge.

Ok. Agreement is good. I appreciate that.

I hope you agree that building a mathematical model of the problem is a valid way to approach it.

I do recognize that "a supertask is not just an infinite sequence," as you've often told me. I agree with that too.

But if you will accept that "0, 1, 0, 1, ..." does not converge," I can work with that for a start.

I disagree with your claim that with respect to Thomson's lamp we can simply stipulate that the lamp is on after two minutes.

I get that. That's where we disagree. I'll try to clarify my argument the best I can.

See my previous post and my initial defence of Thomson on page 13.

Coming up ... at least just your previous post, not necessarily going back through the thread. This is page 24, that's a long time ago. Is there something I should see in there that you haven't said recently?

A supertask is not simply an infinite sequence of numbers.

I agree with that.

In our hypothetical scenario with hypothetical physical laws we are still dealing with the ordinary logic of cause and effect.

I understand your point of view. But here is no cause and effect between the oscillating lamp sequence, and its arbitrarily defined terminal state.

It is implicit in the thought experiment that it is only by pushing the button that the lamp is caused to turn on and off, but strictly speaking this premise isn't necessary as the logic applies regardless of the cause – even if it's magic.

Ok, fine. I'll work with you on that. Let me replace your earlier 60-second model with a 1-second model. That is, there's a final time -- let's call it midnight, in homage to Cinderella's coach. Let's say the light is on at midnight minus 1 second; off at midnight minus 1/2 second; on at midnight minus 1/4 second.

In general, it's on at $\text{midnight} - \frac{1}{2^{n}}$ when n is even, and off when n is odd, for n = 0, 1, 2, 3, ... That is, the lamp is on at the even-exponent inverse powers of 2, and off at the odd-exponent inverse powers of two.

So we have an easy way to know if the lamp is on or off. For example, what is the lamp state at midnight minus 1/128 seconds? Well, 128 is 2 to the 7th power, and 7 is odd, so the lamp is off. So far so good I hope.

If the lamp is on then something caused it to turn on, prior to which it was off. If it is turned on then it stays on until something causes it to turn off.

Well no, this is exactly the spot where you're wrong. Or being more charitable, the spot at which we disagree.

The mathematical model that we've just agreed to (assuming you agree with my notation) defines the state of the lamp at every inverse power of two seconds before midnight.

Say I claim the lamp is on at midnight. If you pick some earlier point, we can determine if the lamp is on or off.

But so what?

Two things are true: The lamp state follows the rule we've given; and the lamp is on at midnight.

That's perfectly fine. There is no contradiction I think you should be much more clear that you think that's impossible.

Likewise, the state at midnight could be off. That too, is perfectly consistent with our lamp state algorithm that works for any point BEFORE midnight.

Given this, if the lamp is on at t1 then either:

a) it was turned and left on prior to t1, or
b) it was turned and left off prior to t1 and then turned on at t1

No it's never "left off" or "left on" prior to midnight. Whatever is its state at $\frac{1}{2^n}$ before midnight, it will have the opposite state at $\frac{1}{2^{n+1}}$ before midnight.

None of that has any bearing whatsoever on the terminal state, which can be anything we like, because, since the on/off sequence has no limit, there is no natural final state.

This "leaving on" or "leaving off" idea is not founded on anything. It doesn't even make sense. It's a consequence of failing to think clearly about the problem.

But as Thomson says, "I did not ever turn it on without at once turning it off ... [and] I never turned it off without at once turning it on", and so both (a) and (b) are false.

So what?

Therefore the lamp is not on at t1. Similar reasoning shows that the lamp is not off at t1 either.

On the contrary. I just showed that it's perfectly sensible for it to be on OR off. Either way is consistent with what's come before; and neither is to be preferred.

I hope you can come even a little ways toward my view. Before midnight, it's an infinite sequence. It has no end. It's never left on or left off. Whatever state it's in, it flips state at the next inverse power of 2.

The basic confusion is not understanding that an infinite sequence has no end. The state at midnight is arbitrary and is not related to what's come before, even under your pretend physics.
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I'm afraid there was a typo in my last post. I posted "Infinity is certainly not a concept", which is rubbish. I meant to post "Infinity is certainly a concept". Apologies.

In online discussions of infinity, someone is bound to come along and say, "Infinity is a concept." Which always makes me say, So What? 3 is a concept. 47 is a concept. Civilization is a concept. The rules of the road on the highway are a concept. Saying infinity is a concept doesn't actually tell us anything about it. I don't even know what it means to say infinity is a concept. Is there anything that isn't a concept?

Yes, I realized that and was hoping to produce a formulation that would allow a more constructive discussion.

I gave it my best shot just now. cc @Michael
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fishfry will speak for himself. But I think the point is that, even a convergent sequence, which does have a limit, does not have a end or last step defined - indeed, is defined as not having one. That means that any answer whatever is equally valid and invalid.

You totally get it. @Michael's turn :-)
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The basic confusion is not understanding that an infinite sequence has no end.

I understand that it has no end. That is why I am arguing that it is metaphysically impossible for an infinite succession of button pushes to end after two minutes.
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