How can a sequence of operations in which each occurs after the other complete without there being a final operation? — Michael

By definition, the sequence completes by having every operation occurring before some finite time. To demonstrate otherwise, one must find a remaining operation which necessarily is not completed at that time.

That there must be a final step in such a sequence not only does not follow from the description of such a supertask, it in fact directly contradicts the description of the supertask.

To demonstrate the impossibility of Zeno's physical supertask, one must attack the premise, not the logic. The logic is valid, at least until he additionally posits the impossibility of the first premise, but that only gives rise to a direct contradiction, not a paradox.

X is a true fact of motion. X is is a false fact of motion. Therefore either motion is impossible, or at least one of the premises is wrong.

By definition, the sequence completes by having every operation occurring before some finite time. — noAxioms

What does it mean for every operation to occur without some final operation occurring?

As it stands your definition is a contradiction.

In my opinion noAxioms isn't right but he was better at the time, and Michael makes a good point, but also isn't right. It's my theory it stands incomplete unless completed through potentiality, in theory it can('t) be done.

What does it mean for every operation to occur without some final operation occurring? — Michael

It means that is isn't a finite sequence of operations. How is it a contradiction that there isn't a final natural number? Instead of just asserting it, show it.

@Barkon also seems to be running on his intuitions and makes unjustified assertions.
If I'm wrong, don't just tell me; show me where.
If Michael is wrong, don't just tell him; show him where.

it's sort of not looked at. You measure one infinity against another stronger infinity that is untop of the first. It 'can' naturally complete this sequence, but in saying that, it can't be shown other than by talking infinities.

It means that is isn't a finite sequence of operations. — noAxioms

No, it doesn't. Saying that it is an infinite sequence of operations means that it isn't a finite sequence of operations.

I'm asking you to make sense of the "every operation is performed" part of "every operation is performed in an infinite sequence of operations”.

I am wrong actually, there is a 'finite' answer. Michael knows what he is on about - ask him.

I found that discussion very helpful. — Ludwig V

Glad to hear that.

But in the staircase problem, if 1 is "walker is on the step" and 0 otherwise, then we have the sequence 1, 1, 1, 1, ... which has the limit 1. So 1, the walker is on the step, is the natural state at the end of the sequence.
— fishfry
Have I understood right, that 0 means "walker is not on the step", and that "the step" means "the step that is relevant at this point" - which could be 10, or 2,436? So 0 would be appropriate if the walker is on the floor from which the staircase starts (up or down)
My instinct would have been to assign 0 also to being on the floor at which the staircase finishes (up or down). It makes the whole thing symmetrical and so more satisfying. — Ludwig V

Could be. Truth be told I got lost in the OP involving many non-relevant fairy tale elements and probably don't even understand what the staircase question is.

That's because the first step backward from any limit ordinal necessarily jumps over all but finitely members of the sequence whose limit it is.
— fishfry
I don't like that way of putting it, at least in the paradoxes. Doesn't the arrow paradox kick in when you set off in the.reverse direction? Or perhaps you are just thinking of the numbers as members of a set, not of what the number might be measuring. I suppose that's what "ordinal" means? — Ludwig V

Ordinals are [ul=https://en.wikipedia.org/wiki/Well-order]well-ordered sets[/url].

As with my standard example, if you take the ordered set <1/2, 3/4, 7/8, ..., 1>, where I'm now using angle brackets to denote ordered sets, suppose you start from 1 and take a step back. Since 1 has no immediate predecessor, any step back necessarily jumps over all but finitely many members of the sequence. It's a counterintutive quirk of limit ordinals. Any path back to the beginning involves only finitely many steps, because the first step back makes such a jump.

Michael's way of putting the point is, IMO, a bit dramatic. — Ludwig V

Yeah. "Metaphysically impossible!" "Non sequitur!" "Nonsense!" Never an actual argument. Tagging @Michael so as not to disparage him behind his back.

The boring truth for me, is that the supertask exists as a result of the way that you think of the task. If you think of it differently, it isn't a supertask. It's not about reality, but about how you apply mathematics to reality. — Ludwig V

I still don't know if walking across the room is a supertask or not.

Not to mention that, if we take the real numbers as a model of space, we pass through uncountably many points in finite time. That's another mystery.
— fishfry
Well, if you insist on describing things in that way .... I'm not sure what you mean by "model". — Ludwig V

Nobody knows the ultimate metaphysical truth about reality. All we can ever do is model is. Relativity is a mathematical model, as is quantum physics, as was Newtonian physics. All science can ever do is build models that fix the experimental data to a reasonable degree of approximation. That's all I mean by model.

I think of what we are doing as applying a process of measuring and counting to space - or not actually to space itself, but to objects in space. — Ludwig V

Only to our latest conceptual model of space. We can't know ultimate reality. Or if we can, we don't as of yet.

A geometrical point has no dimensions at all. So it is easy to see how we can pass infinitely many points in a finite time. (I'm not quite sure how this would apply to numbers, but they do not have any dimensions either.) This doesn't apply to the paradoxes we are considering, which involve measurable lengths, but it may help to think of them differently. — Ludwig V

The unit interval [0,1] has length 1 and is composed of uncountably many zero-length points. That's a mystery.

Name the first one that's not. It's a trivial exercise to identify the exact time at which each natural number is spoken. "1" is spoken at 60, "2" at 90, "3" at 105, "4" at 112.5, and so forth.
I did not "simply assert" all the numbers are spoken. I proved it logically. Induction works in the Peano axioms, I don't even need set theory.
— fishfry
Yes, but you didn't speak all the natural numbers, and indeed, if induction means what I think it means, your argument avoids the need to deal with each natural number in turn and sequence. — Ludwig V

I apparently misunderstood @Michael's backward counting example, I'll be addressing that shortly as I slog through my mentions.

But if I count forward at successively halved intervals: Saying "1" at time 1, "2" at time 1/2, "3" at time 1/4, etc., I will certainly count all the numbers. You can't name the first one I don't say. And we can calculate exactly what time I'll say 47, or Googolplex, or Graham's number.

After 60 seconds I said "0", 30 seconds before that I said "1", 15 seconds before that I said "2", 7.5 seconds before that I said "3", and so on ad infinitum. — Michael

I see that I misunderstood your idea. You are counting time backward. Ok I'll respond to that. But just wondering, when you realized I misunderstood you earlier, why didn't you point that out?

Ok. Suppose that I start at 1 and count backward through 1/2, 1/4, 1/8, ...

Clearly I say all the numbers. at 1 I say 1, at 1/2 I say 2, at 1/8 I say 3, and in general at $\frac{1}{2^n}$ I say n.

It's perfectly clear that I say all the numbers, and iterate through all the negative powers of 2. This is elementary. What number don't I say?

What natural number did I not say? — Michael

There is no natural number that wasn't said. Therefore they were all said.

You can't answer, therefore it is metaphysically possible to have recited the natural numbers in descending order. — Michael

It's perfectly obvious that an infinite sequence is infinite at one end. So you can iterate in one direction and not the other. I can't for the life of me imagine why you think that means anything important.

Now look at the sequence 1, 1/2, 1/4, 1/8, ... again. Graph the points on the real number line. You start at 1, then move leftward to 1/2, then leftware to 1/4, and so on.

The sequence has the well-known limit 0.

Now if you were to start at 0 and move any positive length to the right, no matter how small, you would necessarily jump over all but finitely many elements of the sequence. That's inherent in the meaning and definition of a limit point.

It's exactly the same as 1/2, 3/4, 7/8, ..., whose limit is 1. In fact it's the exact same situation but with the order relation reversed.

Now you want to impose some kind of Newtonian understanding of time, call 0 a time, and say this proves something. It proves nothing but ... and I don't know any other way to say this ... it proves nothing but your own lack of clear thinking around the nature of limits of sequences.

So main points;

* If you iterate through 1, 1/2, 1/4, 1/8, ... while vocalizing "0", "1", "2", "3", etc., you will iterate through ALL the elements of the sequence and you will vocalize ALL of the natural numbers. After all, what member of the sequence do you think is missing? What natural number won't be vocalized?

* Secondly, it's perfectly clear that an infinite sequence starts at one element and continues indefinitely, with no last element. So of course you can't iterate an infinite sequence "from the end." But this is a triviality, it has no significance.

Obviously the above is fallacious. — Michael

Fallacious! Non sequitur! Metaphysically impossible! Nonsense!

These are words. They are not arguments.

It is metaphysically impossible — Michael

There you go again, as Ronald Reagan once said to Jimmy Carter.

to have recited the natural numbers in descending order. — Michael

Sure, because an infinite sequence has no end. You seem to think this elementary and trivial fact has deep meaning. It does not.

The fact that we can sum an infinite series with terms that match the described and implied time intervals is irrelevant. The premise begs the question. And the same is true of your version of the argument. — Michael

I'm disappointed that you won't engage with the argument I'm making. I'll add "begs the question" to your list of buzzphrases used in lieu of substantive argument.

Argument 1
Premise: I said "0", 30 seconds after that I said "1", 15 seconds after that I said "2", 7.5 seconds after that I said "3", and so on ad infinitum.

What natural number did I not recite? There is no answer. Therefore I have recited the natural numbers in ascending order. — Michael

I believe you have agreed with me.

Argument 2
Premise: I said "0", 30 seconds before that I said "1", 15 seconds before that I said "2", 7.5 seconds before that I said "3", and so on ad infinitum.

What natural number did I not recite? There is no answer. Therefore I have recited the natural numbers in descending order. — Michael

No, once again you recited the natural numbers in ascending order.

Normalizing the 60 to 1 for simplicity, you iterated though the sequence 1, 1/2, 1/4, 1/8, ..., while reciting the numbers 1, 2, 3, ... etc.

These arguments only show that if I recite the natural numbers as described then I have recited all the natural numbers, but this does nothing to prove that the antecedent is possible, and it is the possibility of the antecedent that is being discussed. As it stands you're begging the question. — Michael

Begging the question! Nonsense! Non sequitur! Metaphysically impossible!

Always a buzzphrase, never a substantive argument.

Why don't you engage with the argument I'm making?

Now let's assume that it's metaphysically possible to have recited the natural numbers in ascending order and to have recorded this on video/audio. What happens when we replay this video/audio in reverse? — Michael

Lol. You can't play it in reverse, there is no end to the natural numbers in their usual order. You never get to the end. But there is no number you don't vocalize at time point, so you do vocalize them all.

I know this pushes hard against your intuition. If you'll engage with the argument, you will at some point develop better intuitions.

It's the same as having recited the natural numbers in descending order which you admit is metaphysically impossible. Therefore having recited the natural numbers in ascending order must also be metaphysically impossible. — Michael

Completely false, as I've demonstrated numerous time.

Both Argument 1 and Argument 2 are unsound. The premises are necessarily false. It is impossible in principle for us to recite the natural numbers in the manners described. — Michael

Irrelevant! Non sequitur! Metaphysically impossible! Nonsense! Begging the question!

I think I'll just start speaking your language. Maybe that will work.

I had a thread on that a while ago if you care — Lionino

Thanks, I'll check that out. Perhaps it will give me insight into what @Michael means by metaphysically impossible.

Transfinite ordinal numbers are numbers.
Are they? Does √ω have meaning? — noAxioms

5 is a natural number in the Peano axioms. Does $\sqrt 5$ have meaning? No. You have to extend to a larger number system.

$\sqrt \omega$ has no meaning in the ordinals, but I believe it does have meaning in the Surreal numbers, which I don't know much about.

You can't say "x isn't a number because I can't take its square root." You couldn't take the square root of -1 before someone discovered imaginary numbers.

The question of what is a number is historically contingent. Cantor was the one who discovered the ordinals.

It's sad IMO that everyone has heard of the transfinite cardinals, yet nobody knows about the ordinals. The ordinals are logically prior to the cardinals. These days cardinals are actually defined as particular ordinals.

It does for numbers. It's a serious question. I am no expert on how transfinite ordinal numbers are treated. It seems like a different species, like having a set {1, 2, 3, ... , green} which is also a valid set, and countable. — noAxioms

In standard set theory, elements of sets must be other sets. But if you allow urelements, which are elements of sets that are not themselves sets, then you can put green into a set if you like. It's not forbidden by the rules of set theories that have urelements.

https://en.wikipedia.org/wiki/Urelement

But naturals aren't integers which aren't rationals which aren't reals which aren't complex numbers which aren't quaternions. There are lots of different kinds of numbers with different rules, and they were all discovered by the historically contingent work of mathematicians.

Ordering irrelevant. The set supposedly needs to be countable, and it is. Michael's definition of supertask came from wiki, and that definition says it is countable, else it's a hypertask. The SEP definition of supertask omits the 'countable' part and seemingly groups the two categories under one word. — noAxioms

I should read that SEP article, I'd probably get a better understanding of this thread. Wiki giveth and Wiki taketh away. Wiki has many errors.

The definition also includes 'sequential', meaning parallel execution of multiple steps is not allowed. — noAxioms

Hmm, that's interesting. In computer science you can always linearize parallel streams, there's no difference in computational power between parallel and serial processing.

Yes ok but then ... how is walking across the room by first traversing 1/2, then half of the remaining half, etc., not a supertask?
Clearly it isn't a supertask if it is impossible to go only half the remaining distance for some intervals. If that is possible, then it must be a supertask. — noAxioms

Ok, then since walking is commonplace, so are supertasks. I gather @Michael would disagree. I haven't got an opinion.

It violates thebijunction
— noAxioms
I take that back. It doesn't violate the bijection. And I spelled it wrong too. So many errors. — noAxioms

No prob, I figured it out. But there are many many ways to re-order a countably infinite set. Here's one called the even-odd order:

<0, 2, 4, 6, 8, ..., 1, 3, 5, 7, ...>

You can see that this set is still in bijection with the natural numbers, but it's order-isomorphic to two consecutive copies of the naturals. This is a representation of the ordinal $\omega + \omega$.

Note that I no longer have an order-preserving bijection.
That's fine. The rational numbers are both ordered and countable, but they cannot be counted in order. — noAxioms

Yes. Although the rationals don't represent any ordinal. The ordinals only apply to well-ordered sets.

https://en.wikipedia.org/wiki/Well-order

Ah yes, why am I doing all this?

Sounds like the lamp problem is unsolved. It is still 'undefined'. — noAxioms

It's not undefined. Inspired by the story of Cinderella, I defined the terminal lamp state as a plate of spaghetti. I have solved the lamp problem to my satisfaction.

Another note: The paradox of the gods that I occasionally bring up is fun to ponder, but it isn't a supertask since it cannot be completed (or even started). Progress is impossible. Ditto with the grim reaper 'paradox' where I die immediately and cannot complete the task. — noAxioms

So many paradoxes, so little time. I know many philosophers care about these things a lot.

Your ω might help with the stairs. The guy is at 'the bottom' and there is but the one step there, labeled ω. No steps attached to it, but step on that one step and up you go, at some small finite numbered step after any arbitrarily small time. — noAxioms

Right, but unlike the lamp, there IS a naturally preferred solution to the staircase. If the walker is on each step at each time, then defining the walker to be present at the bottom of the stairs preserves the continuity of the path. So the staircase (if I even understood the problem, which I may not have) at least has a natural terminating state. Whereas the lamp definitely doesn't.

Unless the answer is that we satisfy Zeno and execute a supertask every time we walk across the room. But Michael objects to that, for reasons I don't yet understand.
His assertion isn't justified, I agree. — noAxioms

Well I agree with you there, but I can't seem to get @Michael to agree :-)

Some speculative physicists (at least one, I believe) think the world is a large finite grid
So much for the postulates of relativity then. I kind of thought we demolished that idea with some simple examples. It seems to be a 'finite automata' model, and the first postulate of SR is really hard (impossbile) to implement with such a model, so a whole new theory is needed to explain pretty much everything if you're going to posit something like that. I haven't read it of course, so any criticism I voice is a strawman at best. — noAxioms

Finite discrete universe is pretty obscure. I don't know if it's ruled out by other physics or not.

The chessboard universe sounds very classical, and it's been proven that physics is not classical, so I wonder how this model you speak of gets around that. — noAxioms

No idea. Found a physics.SE thread.

https://physics.stackexchange.com/questions/22769/is-the-universe-finite-and-discrete

If supertasks are impossible and motion is possible then motion isn't a supertask.
— Michael
This evaded the question ask. Sure, we all agree that if supertasks are impossible, then supertasks are impossible. He asked how you justify the impossibility of a supertask. All your arguments seem to hinge on a variant that there isn't a largest natural number. — noAxioms

Yay you're helping me gang up on @Michael :-) He and I have been having this conversation.

I think I'll go read the SEP article on supertasks.

√ω has no meaning in the ordinals, but I believe it does have meaning in the Surreal numbers, which I don't know much about. — fishfry

OK. I'll accept that. I do believe somebody has shown no limit to the potential cardinality of some sets.

But naturals aren't integers which aren't rationals which aren't reals which aren't complex numbers which aren't quaternions.

Missed one. :smile:

Wiki has many errors.

Ditto with SEP.

In computer science you can always linearize parallel streams, there's no difference in computational power between parallel and serial processing.

I worked a great deal of my career writing code for multiple processors operating under the same address space. It gets interesting keeping them from collisions, with say two of them trying to write different data to the same location.
Anyway, not sure what you mean by your statement. It seems on the surface to say two processors is no more powerful than one, which isn't true, but two also isn't twice as powerful.

Clearly it isn't a supertask if it is impossible to go only half the remaining distance for some intervals. If that is possible, then it must be a supertask.
— noAxioms
Ok, then since walking is commonplace, so are supertasks.

You didn't read my comment then. Ability to move is a given (an axiom, not something that can be proven). Given that, doing so is a supertask only if Zeno's premise holds, that for any starting point, one must first move halfway to the goal. I can't prove that it holds, but I can't prove that it doesn't hold either.

Yes. Although the rationals don't represent any ordinal. The ordinals only apply to well-ordered sets.

OK. Yet another thing I didn't know.

I defined the terminal lamp state as a plate of spaghetti.

Yes, the PoS solution.

unlike the lamp, there IS a naturally preferred solution to the staircase. If the walker is on each step at each time, then defining the walker to be present at the bottom of the stairs preserves the continuity of the path. So the staircase (if I even understood the problem, which I may not have) at least has a natural terminating state.

Does 'bottom of the stairs' imply a bottom step? If every other step was black and white, what color is the bottom step? PoS, I know. Same problem from where I stand.

No idea. Found a physics.SE thread.

I'll look at that. I have all the respect for the PSE guys, who blow everybody else away. Quora stands somewhat at the opposite end of that spectrum.

It [completing without a last step] means that is isn't a finite sequence of operations.
— noAxioms

No, it doesn't. Saying that it is an infinite sequence of operations means that it isn't a finite sequence of operations. — Michael

Finite means bounded. That means a finite sequence of steps that has a first and last step. An infinite sequence means not (a finite sequence of steps that has a first and last step). It being called 'infinite' literally means that the last step you keep referencing doesn't exist.

I see you have Barkon joining your ranks. I hope you find the company good.

you seem to think the supertask is generating so fast it evades us, in fact we can meet it and persevere at the front of its generation, or even cut it all in one swift equation, S.

Lucky he has got Barkon joining his ranks, you can be confusing noAxioms.

Edit: you wouldn't be able to walk the infinite staircase to the end, but you can definitely conceive the completion of the task using mind.

You can't play it in reverse — fishfry

So you're saying that it's possible to have recited the natural numbers in ascending order and possible to have recorded this on audio but impossible to then replay this audio in reverse? That seems like special pleading. Am I metaphysically incapable of pressing the rewind button?

I believe you have agreed with me. — fishfry

I am presenting two versions of your argument; one in which I have recited the natural numbers in ascending order and one in which I have recited the natural numbers in descending order. I am using the second version to illustrate the flaw in the first version.

No, once again you recited the natural numbers in ascending order. — fishfry

No, I'm reciting them in descending order. I'll repeat it again and highlight to make it clear:

I said "0", 30 seconds before that I said "1", 15 seconds before that I said "2", 7.5 seconds before that I said "3", and so on ad infinitum – e.g. my recitation ends with me saying "3" at 12:00:07.5 then "2" at 12:00:15 then "1" at 12:00:30 and then "0" at 12:01:00.

What natural number did I not recite? There is no answer. Therefore I have recited the natural numbers in descending order.

Notice that even if the conclusion follows from the premise that the argument fails because the premise is necessarily false. It is impossible, even in principle, for me to have recited the natural numbers in the manner described.

Returning to your version of the argument:

I said "0", 30 seconds after that I said "1", 15 seconds after that I said "2", 7.5 seconds after that I said "3", and so on ad infinitum – e.g. my recitation starts with me saying "0" at 12:00:00 then "1" at 12:00:30 then "2" at 12:00:45 and then "3" at 12:00:52.5.

What natural number did I not recite? There is no answer. Therefore I have recited the natural numbers in ascending order.

Even if the conclusion follows from the premise I do not accept that the premise can possibly be true. Like with the previous argument, I think that it's impossible, even in principle, for me to have recited the natural numbers in the manner described.

I have attempted at least to explain why this is impossible (e.g. with reference to recording us doing so and then replaying this recording in reverse), but as it stands you haven't yet explained why this is possible. If you're not trying to argue that it's possible – only that I haven't proved that it's impossible – then that's fine, but if you are trying to argue that it's possible then you have yet to actually do so.

Can you prove that it's metaphysically possible for me to halve the time between each subsequent recitation ad infinitum? It's not something that we can just assume unless proven otherwise. Even Benacerraf in his criticism of Thomson accepted this.

Infinity is said to go on forever, so in some manner we can say it has an everlasting aspect. If one infinity is set such as an infinite staircase, then another is set as walking the infinite staircase, and we have created corelate negative & positive infinites, then in the everlasting realm of both is a perfect conclusion. When both infinites are in their everlasting aspect, at some point there will be change.

Would you then argue that infinity is like exposed electric or wiring and it cannot conclude?

To demonstrate the impossibility of Zeno's physical supertask, one must attack the premise, not the logic. The logic is sound, at least until he additionally posits the impossibility of the first premise, but that only gives rise to a direct contradiction, not a paradox.

X is a true fact of motion. X is is a false fact of motion. Therefore either motion is impossible, or at least one of the premises is wrong. — noAxioms

That's almost right, the logic is valid, but not necessarily sound. Soundness requires true premises. Generally though, judgement of the premises is dependent on empirical knowledge, which all good philosophers know is unreliable. Therefore we have valid logic and if the premises are disproven they would be disproven by competing premises, and the judgement ought not be based on empirical knowledge..

The premise of infinite divisibility is provided from mathematical axioms. The conflicting premise which would be used to disprove this, the limitations of divisibility, cannot be derived from the unreliable empirical knowledge, and it has not yet been provided. So we need to defer judgement, until we can disprove the mathematical axioms in a more reliable way.

I'm going to address Benacerraf's Tasks, Super-Tasks, and the Modern Eleatics:

Thomson's first argument, concerning the lamp, is short, imaginative, and compelling. It appears to demonstrate that "completing a super-task" is a self-contradictory concept. Let me reproduce it here:

_{There are certain reading-lamps that have a button in the base. If the lamp is off and you press the button the lamp goes on, and if the lamp is on and you press the button, the lamp goes off. So if the lamp was originally off and you pressed the button an odd number of times, the lamp is on, and if you pressed the button an even number of times the lamp is off. Suppose now that the lamp is off, and I succeed in pressing the button an infinite number of times, perhaps making one jab in one minute, another jab in the next half minute, and so on. ... After I have completed the whole infinite sequence of jabs, i.e. at the end of the two minutes, is the lamp on or off? ... It cannot be on, because I did not ever turn it on without at once turning it off. It cannot be off, because I did in the first place turn it on, and thereafter I never turned it off without at once turning it on. But the lamp must be either on or off. This is a contradiction.}

Rarely are we presented with an argument so neat and convincing. This one has only one flaw. It is invalid. Let us see why. Consider the following two descriptions:

A. Aladdin starts at t₀ and performs the super-task in question just as Thomson does. Let t₁ be the first instant after he has completed the whole infinite sequence of jabs – the instant about which Thomson asks "Is the lamp on or off?" – and let the lamp be on at t₁.

B. Bernard starts at t₀ and performs the super-task in question (on another lamp) just as Aladdin does, and let Bernard's lamp be off at t₁.

I submit that neither description is self-contradictory, or, more cautiously, that Thomson's argument shows neither description to be self-contradictory (although possibly some other argument might).

The fallacy in his reasoning is that it does not acknowledge that for all t_n >= t_1/2 the lamp is on iff the button was pushed when the lamp was off to turn it on and the lamp is off iff the button was pushed when the lamp was on to turn it off. The lamp "arbitrarily" being on or off at t₁ makes no sense.

We have seen that in each case the arguments were invalid, that they required for their validation the addition of a premise connecting the state of the machine or lamp or what have you at the ωth moment with its state at some previous instant or set of instants. The clearest example is that of the lamp, where we can derive a contradiction only by explicitly assuming as an additional premise that a statement describing the state of the lamp (with respect to being on or off ) after all the switchings is a logical consequence of the statements describing its state during the performance of the super-task.

This logical consequence can be shown when the experiment is explained more clearly:

A. At t₀ the lamp is off, at t_1/2 I press the button

B. At t₀ the lamp is off, at t_1/2 I press the button, at t_3/4 I press the button, at t_7/8 I press the button, and so on ad infinitum

The lamp being on or off at t₁ must be a logical consequence of the lamp being off at t₀ and the button-pressing procedure that occurs between t₀ and t₁ because nothing else controls the behaviour of the lamp.

With (A) we can deduce that the lamp is on at t₁. But what of (B)? If no consistent conclusion can be deduced then its button-pressing procedure is proven metaphysically impossible.

√ω has no meaning in the ordinals, but I believe it does have meaning in the Surreal numbers, which I don't know much about.
— fishfry
OK. I'll accept that. I do believe somebody has shown no limit to the potential cardinality of some sets. — noAxioms

Not sure what you mean by potential cardinality.

I worked a great deal of my career writing code for multiple processors operating under the same address space. It gets interesting keeping them from collisions, with say two of them trying to write different data to the same location. — noAxioms

Point being that you get no increase in computational power from parallelization.

Anyway, not sure what you mean by your statement. It seems on the surface to say two processors is no more powerful than one, which isn't true, but two also isn't twice as powerful. — noAxioms

No function is computable by a parallel process that's not already computable by a linear process. Talking computability theory, not software engineering.

You didn't read my comment then. Ability to move is a given (an axiom, not something that can be proven). — noAxioms

I proved it at the supermarket today, unless you think my vat programmers fooled me again.

Given that, doing so is a supertask only if Zeno's premise holds, that for any starting point, one must first move halfway to the goal. I can't prove that it holds, but I can't prove that it doesn't hold either. — noAxioms

Well maybe it's all an illusion.

I defined the terminal lamp state as a plate of spaghetti.
Yes, the PoS solution. — noAxioms

LOL

Does 'bottom of the stairs' imply a bottom step? If every other step was black and white, what color is the bottom step? PoS, I know. Same problem from where I stand. — noAxioms

Coloring the steps reduces to the lamp.

I'll look at that. I have all the respect for the PSE guys, who blow everybody else away. Quora stands somewhat at the opposite end of that spectrum. — noAxioms

My Quora feed gives me a lot of cute cat pics lately. Makes me happy. Quora certainly used to be a lot better.

You can't play it in reverse
— fishfry

So you're saying that it's possible to have recited the natural numbers in ascending order and possible to have recorded this on audio but impossible to then replay this audio in reverse? That seems like special pleading. Am I metaphysically incapable of pressing the rewind button? — Michael

If you play the recording in reverse, the very first movement of the tape or recording, no matter how small, must necessarily jump over all but finitely many of the vocalizations. For the same reason I've explained earlier. Cute thought experiment though.

But it's just like stepping backward from the limit of a sequence of real numbers. The first step, no matter how small, jumps over all but finitely members of the sequence. It's the same fact as saying that any circle drawn around a limit point necessarily contains all but finitely many elements of the sequence.

I am presenting two versions of your argument; one in which I have recited the natural numbers in ascending order and one in which I have recited the natural numbers in descending order. I am using the second version to illustrate the flaw in the first version. — Michael

I didn't see any flaw. I didn't go back to look up that post, but I do remember responding to it. I can only ask you to reread my earlier response.

No, once again you recited the natural numbers in ascending order.
— fishfry

No, I'm reciting them in descending order. I'll repeat it again and highlight to make it clear:

I said "0", 30 seconds before that I said "1", 15 seconds before that I said "2", 7.5 seconds before that I said "3", and so on ad infinitum – e.g. my recitation ends with me saying "3" at 12:00:07.5 then "2" at 12:00:15 then "1" at 12:00:30 and then "0" at 12:01:00. — Michael

I already responded to this. It's the sequence 1, 1/2, 1/4, 1/8, ..., accompanied by the vocalizations 1, 2, 3, ... Every member of the sequence gets traversed, every natural number gets vocalized.

Since the limit of the sequence is 0, if you start at zero and take even the smallest step forward, you necessarily leap over all but finitely many elements of the sequence.

Do you understand this point? Mathematically I mean, nevermind the element of time, which is a red herring. Do you understand that any interval around the limit point of a sequence must contain all but finitely elements of the sequence? That's the key insight to untangle your example.

Notice that even if the conclusion follows from the premise that the argument fails because the premise is necessarily false. It is impossible, even in principle, for me to have recited the natural numbers in the manner described. — Michael

I've shown several times exactly how to do it, and I've proven that every number gets vocalized.

Even if the conclusion follows from the premise I do not accept that the premise can possibly be true. Like with the previous argument, I think that it's impossible, even in principle, for me to have recited the natural numbers in the manner described. — Michael

I get that you think that. If you would attempt to engage with my argument you might have an insight and develop better intuitions about limits of sequences.

I have attempted at least to explain why this is impossible (e.g. with reference to recording us doing so and then replaying this recording in reverse), but as it stands you haven't yet explained why this is possible. If you're not trying to argue that it's possible – only that I haven't proved that it's impossible – then that's fine, but if you are trying to argue that it's possible then you have yet to actually do so. — Michael

I can't repeat myself again. I have nothing new to say. If you'd read my posts and have yourself a serious think, then come back with a substantive reply, we might get somewhere. We are not making progress.

Can you prove that it's metaphysically possible for me to halve the time between each subsequent recitation ad infinitum? — Michael

That's the premise of your own example. It's not my premise. That's hilarious. In the end, you are reduced to denying your own premise.

It's not something that we can just assume unless proven otherwise. — Michael

It's your example, not mine.

Even Benacerraf in his criticism of Thomson accepted this. — Michael

Feel free to give a reference, else I can't respond.

I already responded to this. It's the sequence 1, 1/2, 1/4, 1/8, ..., accompanied by the vocalizations 1, 2, 3, ... Every member of the sequence gets traversed, every natural number gets vocalized. — fishfry

The fact that there is a bijection between the series of time intervals and the series of natural numbers and that the sum of the series of time intervals is 60 does not prove that the following supertask is metaphysically possible:

I said "0", 30 seconds before that I said "1", 15 seconds before that I said "2", 7.5 seconds before that I said "3", and so on ad infinitum.

How does one start such a supertask?

Feel free to give a reference, else I can't respond. — fishfry

From Tasks, Super-Tasks, and the Modern Eleatics:

What conclusions are we to draw from this rather heady mixture of genies, machines, lamps, and fair and foul numbers? In particular, has it been shown that super-tasks are really possible – that, in Russell's words, they are at most medically and not logically impossible? Of course not. In a part of his paper that I did not discuss, Thomson does a nice job of destroying the arguments of those who claim to prove that super-tasks are logically possible; had there been time I should have examined them. In the preceding section I tried to do the same for Thomson's own neo-Eleatic arguments. I think it should be clear that, just as Thomson did not establish the impossibility of super-tasks by destroying the arguments of their defenders, I did not establish their possibility by destroying his (supposing that I did destroy them).

Also see my comment here where I try to explain where his arguments fail to "destroy" Thomson's.

the OP involving many non-relevant fairy tale elements and probably don't even understand what the staircase question is. — fishfry

There is no bottom, and the OP did not suggest a bottom step. He is done, and no stairs are observable. It's mathematical only, but framed with a physical sounding analogy, which makes it fall apart. — noAxioms

You seem incapable of moving beyond the maths and looking at how you're trying to apply the maths to some proposed real world activity. — Michael

So there is a common understanding of what the issue is. Your disagreement is about different ways of responding to it. Don't you think?

Ryle might have called it a category mistake and talked of putting a physical harness on a mathematical horse or (better, perhaps) putting a mathematical harness on a physical horse, He and many others thought that nothing further needed to be said.

But this problem makes me think that they were wrong. One issue that comes to mind is the issue of making a 2-dimensional map of a 3-dimensional sphere. Euclid doesn't work (accurately). But the problem is resolved by developing a different geometry, which breaks some of Euclid's rules. (I realize I'm oversimplifying here, but I hope I'm not hopelessly mistaken.)

One point to take into account here. This is a thought experiment, so, while the mathematics is real, the horse is not physical, but imaginary, and the difficulty is to work out what rules apply to that in-between context.


I followed the link you gave me and found another link -

Update: https://thephilosophyforum.com/discussion/comment/866916 — Lionino

. This is your exposition of Toulmin's argument about synthetic necessities. Toulmin (for whom I have a lot of time) clearly identifies a class of propositions which orthodox philosophy has not recognized. But he is right.

In order to provide a nutshell explanation, I would say that the point is that the rules of a game can rule out possibilities which are physically possible, but violate the rules of the game. So in a way, they seem to be ruled out a priori or analytically, yet they are physical possibilities. Hence he classifies them as synthetic necessities.

You are interested in exploiting that to define metaphysics. Perhaps that works, perhaps it doesn't. (It's not as if there is any interesting alternative.) But the problem at hand is whether this helps with our problem. I think it does, because it suggests that it is not a matter of discovering what the rules are or what they imply. There is no truth of the matter, because it is a matter of deciding how to apply the rules to a situation which they were not designed to cater for. That's not the same as saying that it is an arbitrary decision, since decisions here may well have consequences elsewhere.

Not sure what you mean by potential cardinality. — fishfry

Pick a number, say 27. I believe it has been shown that there exists a set the cardinality of which is 27, if that's valid terminology. One could also reference aleph-26, but I'm not sure that one can prove that no sets exist with cardinalities between the ones labeled 1 through 27.

Point being that you get no increase in computational power from parallelization.

I beg to differ. A 16 processor machine can sustain a far greater work load than a single-processor machine. The Cray machines were highly parallelized (SIMD architecture) in which thousands of floating point operations were performed by every instruction. These machines were great for stuff like weather simulation.

No function is computable by a parallel process that's not already computable by a linear process.

With that I agree. But that same function can also be done by paper & pencil. You said 'powerful', a reference to how fast the work is completed, and more processors helps with that.

Coloring the steps reduces to the lamp.

I notice that any scenario with a contradiction involves invoking magic. Suppose this physically impossible thing (infinite gods, stairs requiring faster-than-light speed, lamp switches that operate without delay. No magical measurement of something nonexistent. Zeno doesn't do that. No magic invoked, and the first premise thus produces no paradox.

My Quora feed gives me a lot of cute cat pics lately. Makes me happy. Quora certainly used to be a lot better.

Oh it serves its purpose, but correct answers are not promoted above the others, and apparently a great deal of their posters don't know what they're talking about when it comes to stuff like this.

Ryle might have called it a category mistake and talked of putting a physical harness on a mathematical horse or (better, perhaps) putting a mathematical harness on a physical horse, He and many others thought that nothing further needed to be said. — Ludwig V

It is very valid to apply mathematics to physics, but it really helps then if that to which it is being applied is actual physics. Creation of a device to measure a nonexisting thing is not actual physics.

Zeno's horse is quite real. Almost none of the others are.

That's almost right, the logic is valid, but not necessarily sound. — Metaphysician Undercover

Exactly so. I have correct my post. I meant valid and wrote 'sound' in haste. A simple application of modus ponens shows the lack of soundness of Zeno's conclusion iff empirical knowledge is given any weight.

The conflicting premise which would be used to disprove this, the limitations of divisibility

The conflicting premise seemed to be a denial of the completability of a supertask. He never suggests a limit to divisibility.

↪noAxioms you seem to think the supertask is generating so fast it evades us, in fact we can meet it and persevere at the front of its generation, or even cut it all in one swift equation, — Barkon

I have no idea what that collection of words means, so while it may seem to you that I think it, I quite assure you that I don't.

that's nothing distinguishable with any true quota, it's merely your crews signature attitude. Fine, I won't argue against you, but I know most of you are wrong.

It is very valid to apply mathematics to physics, but it really helps then if that to which it is being applied is actual physics. — noAxioms

Very true. I'm afraid what I wrote is a rather embarrassing case of tunnel vision. But it rather matters what mathematics you are trying to apply to what physics. Sometimes it's a case of finding the right mathematics to apply. Which means that it is the physics that's in charge, so to speak.
But that doesn't apply here. Indeed, there's a question whether this branch of mathematics applies to any physics - not that that's an objection to the mathematics itself - just that this isn't the right application of it. That would be a solution, though. (I won't mention the issue of possible future physics'. I don't say it's just arm-waving to discuss it, but it is pretty close.)

Zeno's horse is quite real. Almost none of the others are. — noAxioms

In that case, it is clear what the right mathematics is. (IMO) One of the ways in which Zeno is a better paradox-maker than the others.

Creation of a device to measure a non-existing thing is not actual physics. — noAxioms

If you mean Thompson's lamp, quite so. (Do I understand correctly that Thompson actually argued that supertasks are impossible?) It is a fairy tale which seduces us to look at it wrongly.

I'm sorry I don't know about Zeno's horse — Ludwig V

Look at the context to which my "Zeno's horse" was a reply. You were talking about Ryle saying something on the order of "putting a mathematical harness on a physical horse". It's what Zeno is doing with any of his scenarios, and what almost none of the other scenarios is doing.

If you mean Thompson's lamp, quite so.

The lamp, and almost all the other examples that are not Zeno. They all seem to argue along the lines of <if impossible/self-contradictory thing is true, then contradictions result>. This is a bit like asking "If the sun suddenly didn't exist, how long would it take Earth's orbit to straighten out?"

Do I understand correctly that Thompson actually argued that supertasks are impossible?)

I don't see that. At best he showed that one example is undefined. To prove something impossible it must be shown that there is not a single valid one. To prove them physically possible, one must show only a single case (the proverbial black swan). Nobody has done either of those (not even Zeno), so we are allowed our opinions.

The physical premise of "for something to go from A to B, it must first go halfway there" is very questionable. A great example is a photon going from emitter to detector. Nothing in quantum theory says that the photon is at the halfway point at the time halfway between the emission and absorption events. The principle of counterfactual definiteness (PCD) says it is, and that principle has never been demonstrated to be the case. It is in fact not the case in mosl interpretations of quantum mechanics, including any of the ones that deny faster than light causality.

Look at the context to which my "Zeno's horse" was a reply. — noAxioms

Yes. I realized soon after I had logged off what you were talking about, went back in and edited my response. Too late to avoid revealing how dumb I had been. Never mind, it happens.

At best he showed that one example is undefined......To prove something impossible it must be shown that there is not a single valid one. — noAxioms

That seems to me a good response, though not quite the knock-out blow one would hope for. But it seems to me also a perfectly good reply to a purely mathematical version whether last number is odd or even.
So supertasks are like Gettier problems. Whack one on the head, and another pops up, specifically designed to avoid your refutation. It gets really wearisome, but no-one seems able to find a general refutation. One just gets bored in the end.

I don't see that. At best he showed that one example is undefined. To prove something impossible it must be shown that there is not a single valid one. To prove them physically possible, one must show only a single case (the proverbial black swan). Nobody has done either of those (not even Zeno), so we are allowed our opinions. — noAxioms

Take my explanation of Thomson's lamp above:

A. At t₀ the lamp is off, at t_1/2 I press the button, at t_3/4 I press the button, at t_7/8 I press the button, and so on ad infinitum

Compare with:

B. At t₀ the lamp is off, at t_1/2 I press the button

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.

If no consistent conclusion can be deduced about the lamp at t₁ then there’s something wrong with your button-pressing procedure.

So the fact that the status of the lamp at t₁ is "undefined" given A is the very proof that the supertask described in A is metaphysically impossible. — Michael

The important part is in bold. If there is a problem with the button-pressing procedure, which there is in the case of A, then this problem remains even if the button is broken and doesn't actually turn the lamp on – it turning the lamp on and off isn't the reason that the supertask is impossible but simply demonstrates that the supertask is impossible.

And this problem remains even if rather than press a broken button we recite the natural numbers or even recite a single digit on repeat.

The reason that the supertask in Thomson's lamp is impossible isn't because of what operations are performed but because of how the operations are performed: halving the time between each subsequent operation ad infinitum. This is proven impossible, and as such Thomson's lamp proves that all such supertasks are impossible.

What does it mean for every operation to occur without some final operation occurring? — Michael

A clock ticks 1 time per second.
You start with a cake.
Every second the clock ticks, cut the cake in half.
Make the clock variable, it ticks n times a second.
The limit clock as n tends to infinity applies an infinity of divisions to the cake in 1 second. There is no final operation.

There's nothing logically inconsistent in this, it's just not "physical".

A clock ticks 1 time per second.
You start with a cake.
Every second the clock ticks, cut the cake in half.
Make the clock variable, it ticks n times a second.
The limit clock as n tends to infinity applies an infinity of divisions to the cake in 1 second. There is no final operation.

There's nothing logically inconsistent in this, it's just not "physical". — fdrake

The lamp starts off. Every time the clock ticks a lamp turns from off to on or from on to off as applicable. Thomson's lamp shows that this leads to a logical inconsistency.