Infinite Staircase Paradox
The story:
Icarus was walking through the woods when he stumbled upon a sign which read:
[i]"One, two, three, four, five,
Each step named as down you dive.
Endless stairs stretch out of sight,
Walk them. Grasp the Infinite's might.
Your speed will double with each step as you soar
like an angel approaching a classical black hole's core."[/i]
He glanced down and noticed a dusty staircase plunging into darkness. Thinking that one step would be harmless, he descended and immediately grasped the profound gravity of his actions. He lingered on the first step, marked "1," for 30 seconds, soaking in the enchanting energy coursing through his veins. Moving to step "2," he paused for 15 seconds, feeling lighter and quicker, like a feather in descent. Driven by an irresistible urge, he continued to step "3," then "4,", and so on, each time halving his rest period.
Despite the staircase being endless, he reached the bottom of it in just a minute. Looking around to ascertain his location, he was startled to find himself standing over a dead body. Clearly, the dead body was that of a man who had fallen from a great height. Horrified, he stepped back, intending to ascend the staircase, but it had vanished. Turning around, he found no steps in sight—how could there be, for what would they number?
A wave of anxiety overwhelmed him as he pieced together the events. Slowly, he looked down at the face of the corpse and recognized it as his own.
The infinite staircase was never real—an impossibility and an illusion from the very start. Or was it?
My Questions:
The infinite staircase appears to only allow one to traverse it in one direction. It simultaneously exists and doesn't exist? Does this make sense? If we allow Hilbert's Hotel to exist in the abstract and possible realm, are we forced to accept the infinite staircase into the abstract and possible realm? Is this actually a paradox? What are your thoughts?
Icarus was walking through the woods when he stumbled upon a sign which read:
[i]"One, two, three, four, five,
Each step named as down you dive.
Endless stairs stretch out of sight,
Walk them. Grasp the Infinite's might.
Your speed will double with each step as you soar
like an angel approaching a classical black hole's core."[/i]
He glanced down and noticed a dusty staircase plunging into darkness. Thinking that one step would be harmless, he descended and immediately grasped the profound gravity of his actions. He lingered on the first step, marked "1," for 30 seconds, soaking in the enchanting energy coursing through his veins. Moving to step "2," he paused for 15 seconds, feeling lighter and quicker, like a feather in descent. Driven by an irresistible urge, he continued to step "3," then "4,", and so on, each time halving his rest period.
Despite the staircase being endless, he reached the bottom of it in just a minute. Looking around to ascertain his location, he was startled to find himself standing over a dead body. Clearly, the dead body was that of a man who had fallen from a great height. Horrified, he stepped back, intending to ascend the staircase, but it had vanished. Turning around, he found no steps in sight—how could there be, for what would they number?
A wave of anxiety overwhelmed him as he pieced together the events. Slowly, he looked down at the face of the corpse and recognized it as his own.
The infinite staircase was never real—an impossibility and an illusion from the very start. Or was it?
My Questions:
The infinite staircase appears to only allow one to traverse it in one direction. It simultaneously exists and doesn't exist? Does this make sense? If we allow Hilbert's Hotel to exist in the abstract and possible realm, are we forced to accept the infinite staircase into the abstract and possible realm? Is this actually a paradox? What are your thoughts?
Comments (1106)
The axiom of infinity is how we prove that there is a set that has every natural number as a member.
From the axiom of infinity, we derive that there is a unique least inductive set.
Then we define: w = the least inductive set.
Then we prove that all the natural numbers are members of w.
My point is that we have to be careful in thinking of making this definition:
w = the limit of the sequence of all the natural numbers
since the domain of that sequence is the set of natural numbers, which would already have to have been defined, and so we would have already defined w.
w = the ordinal limit of the sequence of all the natural numbers*
is a theorem, but it would be tricky were it a definition.
* Given a reasonable definition of 'limit of sequence of ordinals'.
/
This is yet another instance of you lashing out against something that I wrote without even giving it a moment of thought, let alone maybe to ask me to explain it more. Your Pavlovian instinct is to lash out at things that you've merely glanced upon without stopping to think that, hey, the other guy might not actually being saying the ridiculous thing you think he's saying. Instead, here you jump to the conclusion that "there's something wrong" with him.
Quite so. Except I thought that it had actually been done.
Quoting fishfry
Quite so. That's why I specified "convergent sequences". (I don't know what the adjective is for sequences like "+1" or I would have included them, because they also have a limit.) "0, 1, ..." is neither. Does the sequent 0, 1, ... have a limit - perhaps the ?th entry?
Quoting TonesInDeepFreeze
Yes. My only point was that it is not a natural number, whereas 1 and 0 are. Hence, although both are limits of their respective sequences, as 1 or 0 also are, 1 and 0 are used in other ways in other contexts. This makes no difference to their role in this context and does not affect their role in other contexts, but does affect what we might call their meaning. ? is not used in any other context - so far as I know.
Quoting fishfry
I agree that we can agree not to ask questions about the lamp outside the context of Thompson's story. But I'm not sure that an assumption really requires a justification. But, for the sake of argument, if I'm telling you a story about a real ball and the shenanigans the prince got up to, you would make that assumption. So if I'm pretending to myself that Cinderella's ball actually happened, I will make the same assumption. This is one reason why I prefer to stick to the abstract structure and shed the dressing up.
Quoting fishfry
Can I ask what your solution is? Just out of interest.
Quoting Metaphysician Undercover
But actions which are outside of the rules are not contrary to the rules, so they are consistent with the rules. However, on thinking about it, I think my answer it that it depends on the rule. Sometimes the rule means that actions that are not permitted are forbidden and sometimes the rule means actions that are not forbidden are permitted. And sometimes neither.
Quoting Michael
Quite so. But how does it help when we are thinking about an infinite sequence? As I understand it, the point is that the sequence cannot define it's own limit. (If it could, it would not be an infinite sequence). The limit has to be something that is not an element of the sequence. It has to be, to put it this way, in a category different from the elements of the sequence. (I'm trying to think of a self-limiting activity, but my imagination fails me. Perhaps later.)
That's precisely the problem. Both of these things are true:
1. The lamp can never spontaneously and without cause be on
2. If the supertask is performed, and if the lamp is on after the performance of the supertask, then the lamp being on after the performance of the supertask is spontaneous and without cause.
Therefore we must accept that the supertask cannot be performed.
And even if we were to grant an alternate account that allows for the lamp to spontaneously and without cause be on, doing so does not answer Thomson's question. He wants to know what the performance of the supertask causes to happen to the lamp. Having some subsequent, independent, spontaneous, acausal event after the performance of the supertask does not tell us what the performance of the supertask causes to happen to the lamp. It's a red herring.
As per P1, the lamp cannot spontaneously and without cause turn into a pumpkin, and there cannot be a god or wizard or gremlin magically turning the lamp into a plate of spaghetti.
And then as per P2, P3, and P4, pushing the button can never cause the lamp to vanish in a puff of smoke.
So the lamp can never turn into a pumpkin. It can never turn into a plate of spaghetti. It can never vanish in a puff of smoke. It can only ever be either off or on.
Before we even consider a supertask, do you at least understand that if the button is pushed to turn the lamp on (and then not pushed again) then the lamp stays on?
Do you at least accept these?
C4. If the button is only ever pushed at 11:00 then the lamp is on at 12:00
C5. If the button is only ever pushed at 11:00 and 11:30 then the lamp is off at 12:00
Quoting fishfry
I did so. It's here.
If the button is never pushed then as per P1 and P4 the lamp will forever be off, consistent with C1.
Quoting TonesInDeepFreeze
Yes, "off" means "not on". The lamp's bulb is either emitting light or not emitting light.
See also the first part of my response to fishfry above.
Not quite. If the last stage of the supertask was on, it is not on spontaneously and without cause.
The problem is that whether the supertasks can be performed is not really the issue. The issue is about how to perform a thought experiment - how much of reality you can import into the story. The state of the lamp, and even its existence, is not defined after the limit. So we can fill in the blanks. You prefer common sense to fantasy, but the story is fantasy, so common sense is not necessarily appropriate.
If you are accepting that the button can be pushed an infinite number of times in a finite time, you have already abandoned causality. I could add a premiss that the lamp cannot switch from one state to another in less than a nano-second, and prove that supertasks cannot be performed. Would that convince anyone? I tried that a long time ago and was put right in short order. You earlier brusquely told me that the reason you didn't run the program was that the computer couldn't process the information fast enough.
A supertask has no last stage. Again to quote Thomson, "I did not ever turn it on without at once turning it off [and] I did in the first place turn it on, and thereafter I never turned it off without at once turning it on."
Therefore, if the lamp is on after having performed the supertask then the lamp being on has nothing to do with me having pushed the button to turn it on. The lamp being on would be spontaneous and without cause, which just isn't possible given our premises.
Quoting Ludwig V
Yes, it is. Thomson's argument attempts to prove that supertasks cannot be performed.
I'm sorry. There is a serious typo in The first sentence of my reply to you. It should have read:-
Quoting Ludwig V
I hope it makes better sense now.
I refer to the last stage only because the question presupposes it. That presupposition is false, of course, which is why there is no answer to the question.
So I don't see the point of your argument here. It's about something else.
Benacerref claimed that the supertask being performed and then the lamp being on is not a contradiction. I am trying to prove that it is (or rather that Thomson already proved this).
The lamp cannot be on after the performance of the supertask and cannot be off after the performance of the supertask – precisely because there is no final button push and because the lamp cannot spontaneously and without cause be either on or off.
The pseudocode I provided before shows this. Its logic does not allow for echo isLampOn to either be determined to output true or false or to arbitrarily output true or false. Therefore, we must accept that it is impossible in principle for while (true) { ... } to ever complete.
And so we must accept that it is impossible in principle for a supertask to be performed.
Thank you. That is much clearer.
If you want to include a wider, more commonsensical context, you could think that a lamp does not spring in to existence at the beginning, or disappear in a puff of smoke after the limit (12:00 or 2:00 or whatever it is). Nor does time stop. But in that case, you can confidently say that its status cannot be determined, with the implication that you need to wait to find out what its status is.
But once you've gone down that road, there are other things you might need to bring in, such as the time it takes for the lamp to transition from one state to another. Then the scenario falls apart - the experiment cannot be conducted.
Quoting Michael
Nor is it. He talks about two instances of the game, and either outcome would be consistent - on its own. But they contradict each other and that's the problem. I don't rate that "refutation" any more than you do.
But, to be fair, he does grant that Thomson's demolition of the arguments for supertasks are valid. It's just his argument against that he takes exception to. It's interesting, though, that neither he nor Thomson considers the other solution - including the limit in the series.
An interesting indeterminate comment. But I think that the impossibility of the final cycle before the limit does put paid to it. It's all about what "complete" means in the context of infinity. Benacerraf, it I've read him right, allows that Achilles can be said to complete infinitely many tasks in a finite time, but argues (rightly) that Thomson's lamp is a different task and suggests to me that he is inclined not to allow that conclusion in that case.
Use of language. When a mathematician says, "X can be done," that's just as good as doing it. There are many jokes around that idea.
There's a formalism or concept called the order topology, in which you can put a topological structure on the set 0, 1, 2, 3, ..., [math]\omega[/math] such that [math]\omega[/math] is a limit point of the sequence, in exactly the same way that 1 is the limit of 1/2, 3/4, 7/8, ...
A topological structure is an abstraction of expressing closeness with open intervals, as in the real numbers. The point is that [math]\omega[/math] is the limit of 0, 1, 2, 3, ... in exactly the same sense as "abstracted freshman calculus," if you think of it that way.
Quoting Ludwig V
No. 0, 1, 0, 1, ... does not have any limit at all. And we can even prove that. Note that it has two subsequences, 0, 0, 0, ... and 1, 1, 1, ,,, that each have respective limits of 0 and 1.
Now it's a theorem that if a sequence converges, all of its subsequences must converge to the same limit. Makes sense, right? A convergent sequence "squishes down" to near the limit.
So a sequence like 0,1,0,1 ... that has two subsequences with different limits, proves that the sequence can not have a limit.
Also, I don't think there even is a name for an arbitrary termination value for a non-convergent infinite sequence. Like
1/2, 3/4, 7/8, ...; 47
In this case 47 is still the value of the "extended sequence" function at [math]\omega[/math]. I call it the terminal state.
I've never seen anyone else use this idea as an example or thing of interest. It doesn't have a name. But to me, it's the perfect way to think about supertasks. The terminal state may or may not be the limit of the sequence; but it's still of interest. It could be a lamp, or a pumpkin, or it could "disappear in a puff of smoke."
Question: Do you put the same constraint on Cinderella's coach? Why or why not? Want to understand your answer.
Regarding the rest of it, I'm lamped out, so I will not debate your ideas further. We have heard each other's talking points multiple times at this point. At least I've got a big time philosopher on my side. How cool is that, right? To actually have professional vindication for a personal idea. I've gotten more than my money's worth from this conversation.
If you feel like answering whether you put the same constraint on Cinderella's coach, I'd be intereted to know. Can't respond anymore to the rest of it. When I get to the point that I haven't typed any words on the subject that I haven't typed before, that's how I know I"m done with that topic.
Thanks for the chat plus any Cinderella comments.
I don't understand your question.
Asking me why I'm using P1 as a premise is as nonsensical as asking me why I'm using P2 as a premise. They are just the premises of the thought experiment. The intention is to not allow for the lamp to be off, for the button to be pushed just once, turning the lamp on – and then for the lamp to be off.
We are trying to understand what it means to perform a supertask, and so we must assert that nothing other than the supertask occurs. There are no spontaneous, uncaused events. If we cannot make sense of what the performance of the supertask (and only the supertask) causes to happen to the lamp then we must accept that the supertask is metaphysically impossible.
I thought that might be your answer. Perhaps we shouldn't pursue the jokes, though.
It's called a performative speech act. Do you know about them? Very roughly, the saying of certain words is the doing. The classic example is promising. A particularly important - and complicated - variety of speech act is a definition. Particularly interesting cases are the definition of rules. (Well, definitions are always regarded as rules, but there are cases that are a bit tricky.)
The relevance is that I'm puzzled about the relationship between defining a sequence such a "+1" and the problem of completion. Each element of the sequence is defined. Done. (And an infinite number of tasks completed, it seems to me). But apparently not dusted, because we then realize that we cannot write down all the elements of the sequence. In addition to the rule, there is a distinct action - applying the rule. That is where, I think, all the difficulties about infinity arise. We understand how to apply the rule in finite situations. But not in infinite situations. Think of applying "countable" or "limit" to "+1". The concept has to be refined for that context, which, we could say, was not covered (envisaged) for the original concept. (By the way, does "bound" in this context mean the same as "limit"? If not, what is the difference?)
Quoting fishfry
Oh, yes, I get it. I think.
Forgive me for my obstinacy, but let me try to explain why I keep going on about it. I regard it as an adapted and extended use of the concept in a new context. (But there are other ways of describing this situation which may be more appropriate.) My difficulties arise from another use of the "1" when we define the converging sequence between 0 and 1. It seems that there must be a connection between the two uses and that this may mean that the sense of "limit" here is different from the sense of ? in its context. In particular, there may be limitations or complications in the sense of "arbitrary" in this context.
Quoting fishfry
I thought so. So when the time runs out, the sequence does not? Perhaps the limit is 42.
Quoting fishfry
So we say that all limited infinite sequences converge on their limits. Believe it or not, that makes sense to me. Since it is also an element of the sequence, it makes sense not to call it a limit.
Quoting fishfry
I have completist tendencies. I try to resist them, but often fail.
That's exactly right. And as I told Michael, way back, in the beginning, if the rules allow for zero importation from the "real world", (which is distinct from the possible world of the supertask), then the allotted amount of time known as the limit, can never pass. And the supertask dilemma never gets off the ground. When Michael insists that the duration must pass, "reality" is needlessly being imported into the thought experiment. We have two incompatible possible worlds.
That the allotted amount of time must pass, if true, is enough evidence to reject the supertask as impossible. However, if we attempt to prove that the amount of time must pass, we run into problems, like those exposed by Hume, namely a lack of necessity in the continuity of time.
I'm glad you agree. And you are right to go on to consider choices we could make.
Quoting Metaphysician Undercover
That's interesting. Do you mean a proof that the amount of time must pass in reality, or a proof that the amount of time must pass in the story? If the former, then we do have a problem. But if the latter, I would argue that the amount of time must pass in order for the conclusion to be drawn. Actually, if the task is suspended before it is concluded for any reason, no conclusion can be drawn either way. So I would think that we have to say that the passing of time is a presupposition of the problem. So I wouldn't use this case as an argument against the infinite divisibility of time (or space, in the case of Achilles). (Actually, following our earlier argument, I'm inclined to see that as a mathematical or conceptual proposition, rather than a fact about the real ("physical") world.)
There's a principle here, that we are willing to import any presuppositions of reality (common sense reality) that are needed to make the argument work, in the sense of drawing a conclusion. But that is limited to what I call presuppositions.
There is another presupposition. There is a presupposition that real people are reading the story and arguing about it - and making choices about how much reality to import.
We can, of course, import whatever we want, in one sense. The issue might then arise whether the new version of the story is the same story or a new one.
It's complicated.
In the thought experiment, the allotted amount of time cannot pass. The switch must complete an infinite (endless) cycle of on/off before the allotted time can pass. The endless cycle cannot ever be finished, so there will always be more switching to do, and the allotted amount of time will never pass. This is just like Zeno's thought experiment, Achilles will never pass the tortoise, because there will always be more distance to cover first..
Now, we add a bit of "reality". Achilles will pass the tortoise, the allotted amount of time will pass. That is reality So we see that what we take for "reality", is inconsistent with, or contradicts what the thought experiment asks us to consider.
We'd think that the rational human being ought to choose "reality" over the ideas of the thought experiment, then we'd reject the nonsense. But this "reality" is concerned with "what will happen", and Hume's problem of induction applies. How do we know that there will be a tomorrow? Because there always has been in the past. How do we know there will be a next hour? Because there always has been. How do we know that there will be a next moment? Because there always has been. However, "because there always has been" does not provide proof that there will continue to be into the future.
Therefore "reality" concerning "what will happen" is lacking in certainty, due to the problem of induction. And, theoretically, a system which prevents the allotted amount of time from passing, through a mechanism similar to the one of the thought experiment could possibly be arranged. Imagine that what is really represented is a continuous slowing down of "our time". Imagine that the mechanism is in a different time frame, so that in the different time frame, the switching on/off is at a constant speed. From our perspective, the switching appears to get faster and faster, but what is really happening is that our time is passing slower and slower in comparison to the other time frame. As it slows more and more and more, it approaches a complete stop, without ever reaching that complete stop, so that every tiny fraction of a second which goes by in our frame, is extremely long in the other. Then it is actually going so slow in comparison to the other time frame, that a very large number of switching can occur in a very short time, and so on as it approaches an infinite amount.
Quoting Metaphysician Undercover
OK. I'm with you that far. Comment:-
That's exactly what Zeno and Thomson want us to do. I guess the complications come in when we want to resist their conclusion, without resorting to "It's just a silly story". We could dismiss lots of perfectly respectable fiction on that basis, of course. But no-one worries about the implausibility/impossibility of the events in "Start Trek" or "Star Wars" or "The Hitchhiker's Guide..". That's where the thought experiment isn't a piece of fiction like a fantasy. Aesop's Fables are also not just a piece of fiction; we are meant to draw conclusions about how to live our lives from them. So "It's just a silly story" is not playing the game. This story wants us to draw a conclusion about how reality is.
Quoting Metaphysician Undercover
Yes. What you are doing is applying the actual context (reality) of the story, but instead of drawing on "common sense", drawing on philosophy. That seems to be not unfair, given that Zeno drew a rather radical philosophical conclusion in direct contradiction with "common sense". (He doesn't even have the grace to compromise by dismissing change as an illusion.) Thomson is different because all he wants to conclude is that supertasks are impossible. That's one thing I've never grasped - If supertasks were possible, what philosophical conclusions would follow?
So, instead of rejecting the idea that time is infinitely divisible, you are turning to Hume and arguing that anything can happen. Maybe you are on stronger ground here. I think some people would feel that you are importing more reality than the rules allow. But I can't be dogmatic about that because I don't really know what the rules are - and I'm certainly not going to argue with Hume - perhaps I'm just shirking a long complicated argument, because I don't think he's right, even though he has a point.
Quoting Metaphysician Undercover
Yes. I don't know how this would play with actual Relativity Theory. But in any case, I don't think that resolves the problem. Why? Because it doesn't actually get Achilles to the finishing line. In the case of Thomson's lamp, it doesn't get to the crunch point when the time runs out. In other words, it postpones, but doesn't resolve, the issue.
For what it's worth, I thought for a while, that one could argue that Achilles always has plenty of time to get to the finishing line, because he passes each stage more and more quickly. But crossing the finishing line requires him to pass the last stage, and there isn't one. That's why I insist that the convergent sequence is not about space or time, but about the analysis of space and time. (I do realize that there's some difficulty understanding that distinction, but I'll assume it for the moment, if I may.) I found something very like that point in Benacerraf's article.
I agree, and the conclusion which needs to be drawn is that there is inconsistency between what we observe as the reality of space and time, and the way that we portray space and time through the application of mathematics.
Quoting Ludwig V
I brought all that in, to say, as I've been saying since the beginning of the thread, that the rejection of the fiction, because it is inconsistent with "reality", isn't that simple. What we know as "reality" has the problem of lacking in certainty. This is what Plato demonstrates in how to deal with Zeno's problems. We cannot simply refer to what we know as "reality", because this is based on sense perception, and the bodily senses are proven to mislead the mind. This is why that sort of "paradox" persists, we dismiss them with reference to "reality" and they go away, that is until skepticism about "reality" reappears.
The issue here is that we really know very little about the nature of the passing of time.
Quoting Ludwig V
The point, was not to resolve the problem, but to demonstrate how the "unreal" situation described could actually be real. So I was showing how the lamp could actually switch on and off infinitely, in the described manner, such that the allotted amount of time would not ever pass, and how it is possible that Achilles might never pass the tortoise. It is an example of time slowing down, and approaching a complete stop. Instead of the action of the lamp switching speeding up, think of the passage of time as slowing down, so it appears like the action is speeding up. Then the point which marks the limit, midnight or whatever never comes
Quoting Ludwig V
I agree with this, but I'd describe it as how we apply mathematics to space and time.
Ok.
Quoting Michael
Pending either of us having anything new to say, I am out of this conversation.
Quoting Michael
"We" does not include me. I regard Thomson's lamp as a solved problem. When you say "there are no spontaneous, uncaused events," you are ignoring the physically impossible premises of the problem. Pushing a button in an arbitrarily small time interval to activate a circuit that likewise switches in an arbitrarily small time interval is already a spontaneous, uncaused event. That's why I commend to you the parable of Cinderella's coach.
Quoting Ludwig V
The jokes illustrate the principle. The mathematicians takes the kettle off the stove and places it on the floor, reducing the problem to one that's already been solved.
Quoting Ludwig V
That tingled the circuit in my memory bank. Searle's doctoral advisor Austin talks about speech acts, and I believe Searle does too. That is everything I know about it. Not really clear what it's about.
Quoting Ludwig V
Well I'm not sure I see what those examples are driving at. Speech where the speech is also an act. So, "It's raining out," is not a speech act, because I haven't done anything, I've only described an existing state of affairs. But telling you how the knight moves in chess (example of a rule] is a speech act, because I've brought the chess knight into existence by stating the rule. Something like that?
Quoting Ludwig V
It's very simple. First, by "+1" do you mean Peano successors? You used this notation several times in what follows and I am not sure I know exactly what you mean.
In Peano arithmetic (PA), we generate all the natural numbers with two rules:
* 0 is a number; and
* If n is a number, then Sn is a number, where S is the successor function.
We can use these two rules to define names like 1 = S0 and 2 = SS0 and so forth, and then use the successor function to define "+" so that we can prove 2 + 3 = 5 and so forth.
There is no "completion" of the sequence thereby generated, 0, 1, 2, 3, 4, ...
In particular, there is no container or set that holds all of them at once. The best we can do is say that there are always enough of them to do any problem that comes up in PA.
That gives you one logical system, PA, that has a certain amount of expressive power. We can do a fair amount of number theory in PA. We can NOT do calculus, define the real numbers, define limits, and so forth.
In PA we have each of the numbers 0, 1, 2, 3, ... but we do not have a set of them. In fact we don't even have the notion of set.
Next step up is set theory, for example ZF, that includes the axiom of infinity. The axiom of infinity actually defines what we mean by a successor function for sets; and says that there is a set that contains the empty set, and if it contains any set X, it also contains the successor of X.
This gives you something PA doesn't: A "container that holds all of 0, 1, 2, 3, ... at once, in fact not just a container, but a set, an object that satisfies all the other axioms of ZF.
We can then show that the axiom of infinity lets us construct a model of PA within ZF; and we take that model to be the natural numbers.
The tl;dr is this:
PA gives you each of 0, 1, 2, 3, ...
ZF with the axiom of infinity gives you {0, 1, 2, 3, ...}; that is, all the marbles AND a bag to put them in.
Hope that wasn't too much information, but it's the way to think of "potential" versus "completed" infinities, which are philosophical terms that don't really find use in math.
Quoting Ludwig V
Mathematical sequences and supertasks are two entirely different, but strongly related, ideas.
There is no time in mathematics. But supertasks are all about time. That's where a lot of the confusion comes in. Supertask discussions talk about time, which is a physical concept; but then examples like Thomson's lamp posit circuits that can change state in arbitrarily short intervals of time, which is a decidedly NON-physical idea. It's a fairy tail (under currently known physics). That's where much of the confusion comes in.
So I hope that you can separate out these two concepts. Are you asking about mathematical sequences, such as 1/2, 1/4, 1/8, ... that have the limit 0? That is a completely understood subject in math.
Or are you imagining that someone "speaks these fractions out loud" in their corresponding amount of time, thereby "saying them all in finite time?" This is a totally nebulous, made-up conceptual fairy tail that is the cause of much confused thinking among philosophers.
Quoting Ludwig V
This is actually not much of an objection. It is far too weak. We cannot write out all the terms of any sufficiently large FINITE sequence, either. You can't write out the numerals from 1 to googolplex in y you lifetime at one number per second. It would take longer than the age of the universe.
So you are not making any substantive objection.
In PA the numbers are conceptually created one at a time, but they're really not, because there is no time. 0 is a number and S0 is a number and SS0 is a number, "all at once." You can call that completion if you like.
In ZF, it's more clear. There is a set that contains 0, 1, 2, 3, ... You can give the set a name and you can work with it.
But either way, your concept of completion involves time; and as I've noticed, that involves CONFUSING mathematical sequences, about which we have perfect logical clarity; with supertasks, about which we have much pretentious confusion.
Quoting Ludwig V
No, that is something you are bringing to the table, but that I don't think is correct. There's no distinct action of applying the rules.
In the PA incantation: 0 is a number and Sn is a number if n is; that creates all the numbers. There is no time involved. Time is a factor that you are letting confuse you.
Quoting Ludwig V
We understand how to apply successors perfectly well in the infinite situation. In fact the rule that "If n is a number, then Sn is a number," is an instance of induction, or its close relative, recursion. These things are perfectly well understood.
Quoting Ludwig V
You are making this up out of some level of confusion involving time. Time is not a consideration or thing in mathematics. All mathematics happens "right here and now."
I am trying, I don't know if I'm getting through or not, but I am trying to get you to separate out your naive notion of timeliness in mathematics, with mathematics. Time matters in physics and in supertask discussions. It's important to distinguish these related but different concepts in your mind.
Quoting Ludwig V
Good question. A bound and a limit are two different things. A couple of examples:
* Consider the set {1/2, 1/4, 1/8, ...}.
-43, that is negative 43, is a lower bound of the set. 0 is the "greatest lower bound," a concept of great importance in calculus.
* But here's a more interesting example. Consider the sequence 1/2, 100, 1/4, 100, 1/8, 100 ...
It has two limit points, 0 and 100. But it has no limit, because the formal definition of a limit is not satisfied. To be a limit the sequence has to not only GET close to its limit, but also STAY close.
0 and 100 would be the greatest lower bound and the least upper bound, respectively.
Now I know this was too much info!! This is just technical jargon in the math biz, don't worry about it two much. But bounds and limits are different concepts. Limits are more strict.
Quoting Ludwig V
Maybe that bit about the order topology was a little too much. My only point is that there is a mathematical context in which omega as the limits of the natural numbers is the same as calculus limits. That's all I need to say about that.
Quoting Ludwig V
This didn't parse, I don't know what you are referring to. What is "it" and "this situation." Nevermind I'll work with the rest of the text.
Quoting Ludwig V
This is a little convoluted and confused. What converging sequence between 0 and 1? Say we have the sequence 1/2, 1/4, 1/8, ... for definiteness.
We can think of this as a FUNCTION that inputs a natural number 1, 2, 3, ... and outputs [math]\frac{1}{2^n}[/math]. I'm starting from 1 rather than 0 for convenience of notation, it doesn't matter.
Now in order to formalize where the limit 0 fits into the scheme of things, we can say that the limit is the value of that function at the point [math]\omega[/math] in the EXTENDED natural numbers
0, 1, 2, 3, ...; [math]\omega[/math]
Those are NOT the natural numbers. I've stuck a conceptual "point at infinity" at the end. I hope this is not confusing you. Tell me what your concerns are.
Quoting Ludwig V
The "termination state" is 42. 42 is not the limit of the sequence 0, 1, 0, 1, ... The word limit has a very technical meaning. It's clear that the sequence does not "get near and stay near" 42.
That's why for purposes of analyzing supertasks I am DEFINING the phrase "termination state" of a sequence to be a value "stuck at the end," but that is NOT NECESSARILY A LIMIT.
I hope this is clear. The termination point is arbitrary, it can be 42 or a pumpkin. But in no case are those values limits in the calculus sense.
Quoting Ludwig V
Hmmm. "Limited" is not a term of art in this context. Given a sequence, it either converges to a limit or it doesn't. A convergent sequence of course converges to its limit, but this is a tautology that follows from the definition of convergence to a limit. A convergent sequence converges to its limit, but that doesn't really any anything we didn't already know.
Quoting Ludwig V
Glad it makes sense, but the limit is NOT repeat NOT part of the sequence.
It's part of what I'm calling the extended sequence, with the limit or termination point stuck at the end. But that is my terminology that I am making up just for these supertask problems.
Hope that's clear.
When I write my semicolon notation: 1/2, 1/4, 1/8, ...; 0
that is a fishfry-defined extended sequence. The sequence is 1/2, 1/4, 1/8, ..., and the limit is 0.
I use this notation to describe the termination state of a supertask: on, off, on, off, ...; pumpkin
The sequence is the on/off part; the pumpkin is the termination state.
Hope this is getting clearer.
Quoting Ludwig V
I don't even know what that means :-) What are completist tendencies?
You would hate the rational numbers then. They are not complete. For example the sequence 1, 1.4, 1.41, 1.412, ... where each term is the next truncation of sqrt(2), does not have a completion in the rationals.
The real numbers are the completion of all the sequences of rationals. That's how we conceptualize the reals.
Well I wrote a lot, let me know if any of this was helpful and let me know what's still troubling you.
tl;dr to this entire post:
Mathematical sequences are clear and rigorous. We have a fully worked out theory of them.
Supertasks are nebulous and vague. Reason: There is no time in math. Time is a concept of physics. And Supertask problems always involve physical impossibilities, like flipping a lamp in arbitrarily small intervals of time. That's the source of all the confusion. Supertasks are fairy tales, like Cinderella's coach; and you can no more apply logic to a supertask problem than you can to the coach turning into a pumpkin.
I traced back to your mention of the axiom of infinity, and I still fail to see the relevance of the remark in context. I apologize for lashing out regardless. "wut" is a standard Internet location, and though it carries a bit of snarkitude, it's not considered overly aggressive in the scheme of things. Just an expression of puzzlement.
wut?
Quoting fishfry
My response was to 'what's wrong with you tonight?', not so much to 'wut?'.
Convenient for you now to self-justify by highlighting 'wut?' and not 'what's wrong with you tonight?'.
There was nothing wrong with what I posted that night. You just snapped-at as if there were, when actually the problem is that you, as often, reply to your careless mis-impression of what is written rather than to what is actually written.
Hey, I get your whole "Aw shucks, I'm just a scorpion who's gonna do what a scorpion's gonna do. I don't mean nothin' by it" routine. But it doesn't mean jack to me as far as feeling any less right in answering right back.
Ah. The what is wrong with you and not the wut. I can see that now that you mention it.
I am terribly sorry to have offended you once again.
You first claimed that I was offensive to you. So I pointed out that you don't realize how offensive you often are. So I just gave you that info. I don't sweat being offended in posts. But you carelessly misconstrue what I've posted, and claim I've said things I haven't said, and write back criticism of my remarks by skipping their substance and exact points. And that is what I post my objections to.
Meanwhile, what you say about my posting style is rot. You say it's too long. But you also say it doesn't explain enough. Can't have it both ways. And I do explain a ton. But, again, I can't fully explain without having the prior context back to chapter 1 in a text already common in the discussion. And l explain somewhat technically because being very much less technical threatens being not accurate enough. Meanwhile, your own posts are usually plenty long, so take that tu quoque.
I've not gone back to review all that's been said in this thread, and I need to catch up to your replies, but starting again from the beginning of your argument.
I surmise that the reason you put your argument in numbered steps is so that it can be seen to be airtight.
Is your argument intended to be Thomson's argument?
You have mentioned different conclusions you draw:
(1) The conditions (the premises) of the lamp are inconsistent.
(2) Supertasks are impossible. (But can we infer from the impossibility of Thomson's lamp that all supertasks are impossible?)
(3) Time is not continuous. (I've suggested that what you actually seem to dispute is that time is not densely ordered (infinitely divisible), which is a stronger claim.)
(4) Benacerraf is wrong.
Here's Thomson's statement of the problem:
"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,
according to Russell's recipe. 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 seems impossible to answer this question.
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."
Here's your presentation:
Quoting Michael
I want to get back to looking at this more closely, but in the meantime, do you consider your presentation equivalent with Thomson's statement of the problem?
When possibility is part of the analysis, the analysis can get complicated. We should be careful that our inferences regarding the modalitiy are proper.
Are you just committed to picking fights with me? I've apologized several times tonight, for sins real and imagined. And some cosines too. Enough bro'.
I won't argue with that. For some reason, I've never been able to get my philosophical head around that topic. Just like Augustine, all that time (!) ago.
Quoting Metaphysician Undercover
I was going to reply that slowing down isn't stopping. I didn't realize that the slowing down was a convergent series. Perhaps slowing down can be stopping.
Quoting Metaphysician Undercover
Well, we could if we wanted to do. But why would we want to? Apart from the fun of the paradox. Mind you, I have a peculiar view of paradoxes. I think of them as quirks in the system, which are perfectly real and which we have to navigate round, rather than resolve. Think of the paradoxes of self-reference. Never permanently settled. New variants cropping up.
I don't ask for apologies. But it's okay if you want to give them. But you embed into your apologies yet more items that I feel deserve response. Your apologies themselves are snarky; "sins imagined" e.g. I don't even object to snark, except it's your way of ostensibly apologizing while still turning it back on me.
If I misconstrue someone's math or philosophy points, especially to mischaracterize them, then if the person calls me on it or I discover it myself, before posting back to that person again, I should post my recognition of my mistake. That's my ethos. Yours might be different. But I will stick with my prerogative to reply when I like.
And to answer your question: No, I definitely do not have any interest in "picking fights" and I find no value in fighting for the sake of fighting. But I do find value in posting disagreements and corrections, whether regarding the math and philosophy or regarding the personal specifics of the posting interchanges. In various thread, you have posted a lot of inaccuracies and misconceptions about math, and now lately about me. I respond to that.
Ok no more snark.
Interesting. I hope I didn't bury the lede. I'm not all up about sarcasm. Rather, what I find important is (1) striving not to misrepresent a poster's remarks and to stand corrected when it is pointed out that one has; and (2) not to argue by ignoring key counter-arguments and explanations; not to just keep replying with the same argument as if the other guy hadn't just rebutted it.
Yes.
No I'm not. I accept that one of the premises of the thought experiment is physically impossible. That doesn't then mean that we cannot have another premise such as "there are no spontaneous, uncaused events".
You seem to think that because we allow for one physical impossibility then anything goes. That is not how thought experiments work.
It is physically impossible for me to push a button 10[sup]100[sup]100[/sup][/sup] times within one minute, but given the premises of the thought experiment it deductively follows that the lamp will be off after doing so. Your claim that the lamp can turn into a plate of spaghetti is incorrect.
In many cases of common language usage, "slowing down" is stopping, but that implies the end, not yet achieved. The point is that "stopping" is distinct from "stopped'. And if the slowing down never reaches the point of being stopped, then the term "stopping" is not justified. The convergent series is misrepresented as "stopping", because the end of "stopped is never achieved.
In the modern physical world of relativity, "stopped" is arbitrarily assigned according to an inertial reference frame. This implies a sort of equilibrium, or stability within that specific reference frame, but it's highly unlikely that it is a true case of "stopped", more likely very slow movement, misrepresented as "stopped". We like to round things off.
Quoting Ludwig V
Why would we want to? Because we are philosophers seeking knowledge. Understanding is the primary objective. I look at such paradoxes as indications of a lack of understanding. The principles applied do not adequately map to the reality which they are being applied, this is a failure of our knowledge. Then we need to subject all the principles to skeptical doubt, to determine the various problems. We could just live with quirks in the system, but that's unphilosophical. Knowledge evolves, and that evolution is caused by people attempting to work out the quirks in the system.
I am so appreciative that you straightened me out on this extensionality thing that I can't argue with you about anything. I accept all your criticisms. You say I've done these things and I don't deny them. I make no defense nor explanation.
I do have a sarcasm gene and that rarely works online. You'd think I'd learn.
I respectfully leave this conversation. We've said it all. i've enjoyed our chat.
Yes I see what you meant. Thanks.
Then you can't argue with me that you can argue with me.
It's very helpful, so that's fine. I get my revenge in this post.
The system is not helping me here, because it invites me to link to specific comments, but I'll do my best to make clear what I'm responding to.
Quoting fishfry
:grin:
Perhaps that's why philosophers keep tripping up on them. It is well known that they don't notice what's on the floor - too busy worrying about all the infinite staircases and the fall of man.
Quoting fishfry
That was not a very well thought out remark. I would certainly have hated them in the long-ago days when the Pythagoreans kept the facts secret so that they could sort it out before everyone's faith in mathematics was blown apart. But now that mathematicians have slapped a label on these numbers and proved that they cannot be completed, I'm perfectly happy with them.
Quoting fishfry
Yes. Austin invented them, Grice took them up, Searle was the most prominent exponent for a long time, although he has now moved on to other things now. It's a thing in philosophy For me, it's a useful tactical approach, but a complete rabbit-hole as a topic.
Quoting fishfry
Something like that. The initial point was to establish that there are perfectly meaningful uses of language that are not propositions (i.e. capable of being true or false), in the context of Logical Positivism. I doubt that you would welcome a lot of detail, but that idea (especially the case of the knight in chess) will be at the bottom of some of the later stuff.
Quoting fishfry
It was very helpful to me. I have doubts about the terminology "potential" vs "completed", but the idea is fine. I particularly liked "don't really find a use in math".
Quoting fishfry
Too much or not. It helped me. Someone else started talking about bounds and I couldn't understand it at all. I may not understand perfectly, but I think I understand enough.
Quoting fishfry
I know that. It's not a problem. If I said anything to suggest otherwise, I made a mistake. Sorry.
Quoting fishfry
... because "1/2, 1/4, 1/8, .." gets near and stays near 0. Yes?
Quoting fishfry
I understand that distinction.
Quoting fishfry
Quoting fishfry
Many of my notions are naive or mistaken. But this separation is my default position. I'm not making an objection, but am trying to point out what may be a puzzle, which you may be able to resolve. On the other hand, this may not be a mathematical problem at all.
Quoting fishfry
There are other ways of putting the point. What about "Mathematics is always already true"? Or mathematics is outside time? Or time is inapplicable to mathematics?
But your example of making a rule in chess. Note that as soon as the rules are made, we can starting defining possibilities in chess, or calculating the number of possible games and so forth. It's as if a whole structure springs into being as we utter the words. So a timeless structure is created by our action, which takes place in time. Isn't that at least somewhat like a definition in mathematics? And the definition is an action that takes place in space and time.
More difficult are various commonplace ways of talking about mathematics.
Quoting fishfry
Quoting fishfry
Quoting fishfry
Quoting fishfry
Quoting fishfry
At first sight, these seem to presuppose time (and even, perhaps space) Perhaps they are all metaphors and there are different ways of expressing them that are not metaphorical. Is that the case? I recognize that I may be talking nonsense.
Quoting Deleted user
So when you use the appropriate sense of the "world", and say that realism is true of the world, you are saying that realism is true of some parts of the world - the abstract parts?
I strongly approve of defining the context in which one is using "real" or "realism", but using it of the world, defined as everything that exists independently of the mind, you are simply re-asserting the basic thesis that both geometries are true independently of the mind. Since they are both true in the abstract world, but not simultaneously in the physical world, would it not be helpful to add that explanation?
Yes, I agree with that. I was suggesting that a slowing down according to a convergent series might count as stopped, since it would never reach the limit or "0".
Quoting Metaphysician Undercover
If you are right about relativity, I wouldn't disagree.
Some time ago I mentioned time dilation in relativity theory in this regard.
Correct. Which is why I acknowledged your complaints and said nothing else. If I did, you'd complain that I was minimizing my apology by contextualizing it, either with snark or denial.
So I didn't even apologize. I acknowledge your complaints and I stand mute. I have nothing to say at all.
Glad to know. Revenge? What do you mean? By writing a long post? Well I write long posts but prefer when others write shorter ones. I haven't solved this dilemma yet.
Quoting Ludwig V
Not sure what you mean. I generally quote the whole post then stick in quote tags around the specific chunks of text I want to respond do.
Quoting Ludwig V
I don't know many philosopher jokes.
Quoting Ludwig V
Sorry maybe I was off track about the rationals.
Quoting Ludwig V
Yes he got in trouble for harassing his female doctoral students.
Quoting Ludwig V
Ok. Why did you bring it up relative to math? Oh I remember. "Let x = 3" brings a variable x into existence, with the value 3. So statements in math are speech acts, in the sense that they bring other mathematical objects into existence. I can see that.
Quoting Ludwig V
Ok. Not entirely sure where you're going.
Quoting Ludwig V
Ok I was only trying to be philosophical. Aristotle (I think) made the distinction. It doesn't come up in math, nobody ever uses the terminology. But the way I understand it is that Peano arithmetic is potential and the axiom of infinity gives you a completed infinity.
Quoting Ludwig V
Ok, bounds. They're just the shoulders of the road. Thing's you can't go past. Guardrails.
Quoting Ludwig V
That was about limits versus "termination state." I should emphasize that limits are perfectly standard mathematical terminology. But "termination state" is my own locution for purposes of talking about supertasks. The termination state is like a limit in the sense that we can conceptually "stick it at the end" of an infinite sequence; it just doesn't have to satisfy the definition of the limit of a sequence. Like 1/2,/ 3/4, 7/8, ...; 42
The semicolon notation is my own too. I don't think mathematicians talk about supertasks. They're more of a computer science and philosophy thing.
Quoting Ludwig V
I am not aware of what problem or puzzle you are expressing.
Quoting Ludwig V
The subject matter of mathematics does not speak about time. That's different than saying "math is outside of time," although it's kind of related. Physics talks about time, and physicists use math to model time, but that is a very different thing.
It's the difference between a loop in math versus programming.
In math when we say that 1/2 + 1/4 + 1/8 + ... = 1, we mean "right now," though even that is a reference to timeliness. The equality "just is."
But in a programming language when we write a loop that keeps adding each term to a running total, that notation stands for a physical process that takes place in a computing device and requires time and energy to execute, and produces heat. A programming loop is a notation for a physical process.
Quoting Ludwig V
Ok. So when I write down the rules of set theory, I instantiate or create all the complex world of sets as studied by set theorists. And you speculate that this might be an event that takes place in time.
There is another point of view. The structures of the sets were there. Mathematicians discovered the structures. So the discovery of set theory is historically contingent and takes place in time, around 1874 or so with Cantor's first paper on set theory. But the sets themselves, the structures of set theory, are eternal!
In other words this is the old "invented or discovered" question of mathematical philosophy.
Now chess, I think we can agree, was invented and not discovered. But math is somehow different. Math is somehow wired into the logic centers of our minds, and perhaps the universe.
Quoting Ludwig V
Are you referring to what I just talked about?
Quoting Ludwig V
I cannot fathom what you might mean. A sequence does not approach its limit in time. The limit of 1/2, 1/4, 1/8, ...is 0 right now and for all eternity. The fact is inherent in the axioms of set theory, along with the usual constructions and definitions of the real numbers and calculus. In that sense the fact "came into existence" when Newton thought about it, or maybe when Cauchy formalized it, and so forth.
But the history of our understanding of the fact is not the same as the fact itself. The earth went around the sun even before Copernicus had that clever idea. Likewise every convergent sequence always converged to its limit, independently of our discovery of those limits, and our understanding of what a limit is.
Is this your point of contention or concern? That you think that time is hiding in there somewhere? I profoundly disagree. You greatly misunderstand mathematics; or you have an interesting and original philosophy of mathematics; if you believe there's time hiding inside mathematics.
Quoting Ludwig V
If I am understanding you, you think time is somehow sneakily inherent in math even though I deny it.
Have I got that right?
It was merely a math quip.
Quoting jgill
I either skimmed past it or forgot it. Sorry. Not having been trained for it, I wouldn't want to comment on it. But it is that left field plausibility that I always appreciate.
Yes. I was saying in a complicated way, that a long post is not, for me, a bad thing.
Quoting fishfry
That's a useful tactic. I shall use it in future.
Quoting fishfry
He did indeed. It was very common back in the day. It was disapproved of by many, but not treated as unacceptable. I don't think anyone can really understand how horrible it is unless they've actually experienced it.
Quoting fishfry
Exactly. There's a lot of refinement needed. But that's the basic idea. What those objects are is defined entirely by their use in mathematics.
Quoting fishfry
I was just being pedantic. It was a thing in the era before Descartes &c. But I understood that the distinction was "potential" and "actual". Nonetheless, the idea of a "completed" infinity catches something important.
Quoting fishfry
That's a very helpful metaphor.
Quoting fishfry
Yes.
Quoting fishfry
Nor can I. That's the problem.
Quoting fishfry
That's the starting-point.
Quoting fishfry
Why is this a problem? The traditional view is that mathematics, as timeless, cannot change. Our knowledge of it can, but not the subject matter. (Strictly that rules out creating any mathematical objects as well, but let's skate over that.) "A sequence does not approach its limit in time" makes no sense.
I may be about to solve my own problem. That doesn't mean that raising it with you is not helpful.
We have to accept that a sequence approaching its limit is not like a train approaching a station. The train is approaching in space and time. But you can't ask what time the sequence left its origin and when it will arrive at its limit.
You can call the sequence approaching its limit a metaphor or an extended use. The train approaching the station is the "core" or "paradigm" or "literal" use. The sequence approaching its limit is a different context, which, on the case of it, makes no sense. So we call this use is extended or metaphorical.
We can explain the metaphor by drawing a graph or writing down some numbers and pointing out that the different between n and the limit is less than the difference between n+1 and the limit is less and that the difference between n and n-1 is greater.
And so on.
Quoting fishfry
Yes. I realize this is border country. Godel seems to live there too.
Quoting fishfry
If PA here is first order, then PA does not have a predicate 'is a number' nor those axioms.
Quoting fishfry
Just to be clear, that occurs in set theory, not in PA. In PA, '+' is not defined. It is primitive.
Quoting fishfry
Of course, that's correct regarding PA.
Quoting fishfry
Right.
Quoting fishfry
Right.
Quoting fishfry
The axiom of infinity does not define anything, including the successor operation.
The successor operation only requires pairing and union:
Df. the successor of x = xu{x}.
That is logically prior to the axiom of infinity. Then the axiom of infinity only says that there is a set that has 0 and is closed under successor.
Then we prove that there is a unique set that is a subset of all sets that have 0 and are closed under successor.
Then we define w = the set that is a subset of all sets that have 0 and are closed under successor.
Quoting fishfry
Not "and". All it says is what you said after the "and": "there is a set that contains the empty set, and if it contains any set X, it also contains the successor of X".
Quoting fishfry
Rather than "the model" I would say "the standard model". There are other models too. And models not isomorphic with the standard model.
Quoting fishfry
Quoting fishfry
Both are right, and well said. In both PA and Z without infinity (even in Z with the axiom of infinity replaced by the negation of the axiom of infinity), we can define each number natural number, and in Z we can prove the existence of the set of all and only the natural numbers.
Oh I see. I prefer shorter posts so I don't get lost in the quoting!
Quoting Ludwig V
Yes, I just highlight the whole post and say Quote.
Quoting Ludwig V
"Back then" wasn't that long ago, this scandal's just a few years old IIRC. I didn't follow the particulars. I'm sure it's just as common today. Or maybe not as 1950's, say. Thinks have changed since then. Still. The male-female thing, those are very deep energies being played with. You are not going to stamp it out with rules. You can change the form in which the scenarios are played out. What Eric Berne called the games. Games People Play, remember that?
Quoting Ludwig V
There's another funny thing that goes on. Sometimes you make a definition, and it DOESN'T bring a mathematical object into existence. For example we write down the axioms for a group. But we have no idea if there are any groups, or if the axioms are perhaps vacuous. So the next step is to exhibit some groups, like the additive structure on the integers. This is a very common pattern in math: Make a definition, then show that there's something that satisfies the definition!
Quoting Ludwig V
Right, potential/actual versus potential/completed. I've heard them both. Since they don't come up in math I kind of use them interchangeably. But they probably have more specific technical meanings or contexts I don't know about. But for me, I just regard the axiom of infinity as the sharp boundary between the two concepts. Induction on the one hand, versus a "completed loop."
Quoting Ludwig V
Shoulders of road. Bounds. Glad that helped.
Quoting Ludwig V
I quoted that entire exchange. I can't fathom your meaning. But you say you can't either.
Time is a concept in physics. You can see that, right? Math is outside of time. It doesn't describe or talk about time, though it can be used by physicists to model time. And then again, we have no evidence that the mathematical real numbers are even a decent model for time. The real numbers are continuous, but nobody knows if time is.
If you can give an example of what you are thinking, that might be helpful.
Quoting Ludwig V
Ok. You seem to be agreeing with me.
Quoting Ludwig V
And now you're not agreeing. The word "approach" is colloquial. It is not intended to evoke images of panthers stalking their prey, or arriving at your destination in a car. Not at all. It's just the word we use for the limiting process. But 1/2 + 1/4 + 1/8 + ... IS 0; it does not "become" or "approach" zero. It's this language ambiguity that is the source of so much online confusion about the subject. See any .999... = 1 debate. You'll hear that .999... "approaches but does not reach" 1. But the sum is exactly 1 nonetheless, by virtue of the definitions. The math is designed to make it work out.
Quoting Ludwig V
I'd like to know what the problem is, regardless. If for no other reason than to make sure you're understanding it correctly!
Quoting Ludwig V
Correct. It's a shame we use the word "approach," because many are confused by that.
Quoting Ludwig V
Right. Trains are physical objects. Numbers in a sequence are mathematical abstractions. They don't live in the physical world.
Quoting Ludwig V
I think it's just a confusing use. When mathematicians use the word approach, in their minds they already have the full context of the theory of limits. So they are not confused. But non-mathematicians hear the word and associate it with is everyday context. No harm is done, till these misunderstanders show up to .999... = 1 threads online. Then we get problems.
Quoting Ludwig V
With supertasks? I don't think so.
I would organize Thomson's argument differently from the way he organizes it. Near the end of his argument, he says "But the lamp must be either on or off." But he's actually invoking a premise. It is natural to regard the lamp as being either Off or On and not both, but in this highly hypothetical context, it would be good to say that as an explicit premise.
Then Thompson invokes infinitely divisible time. But not as a premise. I would include it as a premise. The advantage of doing that is that then the premise is explicitly a candidate for rejection to avoid the contradiction.
I simplified the language of your conclusions (we don't need all those tn, ti, tj and inequality symbols), and I don't think you need the conclusions to be biconditionals to derive that the lamp is neither Off nor On.
('r' for 'revised')
Premises:
rP1: At all times, the lamp is either Off or On and not both.
rP2: The lamp does not change from Off to On, or from On to Off, except by pushing the button.*
*The pushing of the button and the change are together instantaneous, and the button can be pushed only once in any moment. This is not needed except to simplify the argument (especially to state rC1, rC2 and rP6).
rP3: If the lamp is Off and then the button is pushed, then the lamp turns On.
rP4. If the lamp is On and then the button is pushed, then the lamp turns Off.
rP5: The lamp is Off at 10:00.
Conclusions:
rC1: If the lamp is On at a time T2 after 10:00, then it was pushed On at some time T1 that is both after 10:00 and before or at T2, and not pushed at any time that is both after T1 and before or at T2.*
*Notice that T1 and T2 are in chronological order.
rC2: If the lamp is On at a time T1 after 10:00 then Off at a time T3 after T1, then it was pushed Off at some time T2 both after T1 and before or at T3, and not pushed at any time that is both after T2 and before or at T3.*
*Notice that T1, T2 and T3 are in chronological order.
Premise:
rP6: At 11:00 the button is pushed to turn the lamp On, at 11:30 Off, at 11:45 On, and alternating in that way ad infinitum.*
* We could easily make that mathematically rigorous.
Conclusion:
rC3: The lamp is neither Off nor On at 12:00. Contradicts rP1.
QUESTION: How do you state the arguments for rC1 and rC2 from the premises rP1-rP5?
I think that this is what the so-called "paradox" of supertasks is all about. What is revealed is that at least one or the other, space or time, or both, must not be continuous. I think that's what @Michael has been arguing since the beginning. Tones attempted to hide this behind sophistry by replacing the continuity of the real numbers with the density of the rational numbers.
The real issue is that if one of these, space or time, is not continuous, then it cannot be modeled as one thing. There must be something else, a duality, which provides for the separations, or boundaries. But I don't think anyone has shown evidence of such a duality, so we have no real principles to base a non-continuous ordering system on.
Quoting fishfry
I'd say this is similar to Tones' use of "identity" in set theory. We take a word, such as "approach", which clearly does not mean achieving the stated goal, and through practise we allow vagueness (to use Peirce's word), then the meaning becomes twisted, and the use of the word in practise gets reflected back onto the theory. So we have the theory stating one thing, and practise stating something different, then the meaning of the words in the theory get twisted to match the practise. Practise says .999... is equal to 1, so "approach" in the theory then takes on the meaning of "equal". Practise says that two equal sets are identical, so "equal" in the theory takes on the meaning of "identical". These are examples of how theory gets corrupted through practise when the words are not well defined.
There is no sophistry on my part. And no "replacing". I merely pointed out that proving that time is not continuous does not prove that time is not densely ordered (or infinitely divisible).
And the crank is so ignorant and mixed up about this very thread that he wildly infers that my remarks about the thought experiment vis-a-vis Michael's version of it entail that I have myself made certain claims about time beyond that not-continuous does not imply not-dense.
The garbage posting crank doesn't know what he's talking about, regarding continuity or denseness, or me. He is a bane upon reasoned inquiry.
/
The crank is on about .9... Whatever he's trying to say, in his usual thought salad way, we should at least recognize that the notation '.9...' is informal for the limit of a certain sequence.
Meanwhile he has his own utterly mixed up notions about what 'identical' and 'equal' mean. But he hasn't the least reasonability even to understand that his own having notions about what words should mean doesn't entail that everyone else is wrong for using the words both in their ordinary English senses and also in stipulated mathematical senses. He does not understand even the notion of stipulative definition, just as, in another instance, he does not understand even the difference between use and mention.
The crank falsely rails on and on against mathematics and modern logic, even though he has not read page one in a textbook in the subject. As he serves as a textbook example in crank sophistry.
I don't say that selecting and organizing the quotations is easy. It fits better with the fact that I tend to get slabs of time when I can pursue these discussions but in between, I'm not available at all. So the quick back and to is more difficult for me.
Quoting fishfry
I didn't mean to imply that they were living together. That would be .... interestingly mnd-boggling.
Quoting fishfry
Don't get me started. What particularly annoys me is that so many people seem absolutely certain that they are right about that. I think it is just a result of thinking that you can write probability = 1, when 1 means that p cannot be assigned a probability, since it is true. A friend once conceded to me that it was a degenerate sense of probability, which is like saying that cheese is a degenerate form of milk.
Quoting fishfry
Since my earlier comment on this,
Quoting Ludwig V
I've discovered that potential infinity is the definition of the sequence and actual infinity is the completion of the sequence. So "potential" and "completed" can be fitted together after all.
Quoting fishfry
I think I shall stick to my view that defining an infinite sequence or getting a beer from the fridge is the completion of an infinite number of tasks. I don't think it gives any real basis for thinking that supertasks are possible.
Quoting fishfry
You notice that maths outside time is metaphorical, right? I prefer to say that time does not apply to maths, meaning that the grammatical tenses (past, present and future) do not apply to the statements of mathematics. I like "always already" for this. There is a use of language that corresponds to this - the "timeless present". "One plus one is two" makes sense, but "One plus one was two" and "One plus one will be two" don't.
Quoting fishfry
Yes. But there are complications. How does math apply to the physical world?
Quoting fishfry
We have a choice between insisting that Non-Euclidean geometries are not created but discovered and insisting that they are not discovered but created - though they exist, presumably, forever. But if we create them, what happens if and when we forget them?
Quoting fishfry
As I said before there are a number of ways to describe this. They're all a bit weird.
Quoting fishfry
It sounds as if you are saying that "approach" is a simply two different senses of the same word, like "bank" as in rivers and "bank" as in financial institutions. An old word given a new definition. Perhaps.
Quoting fishfry
That's a very neat definition. I'll remember that. But you can see, surely, how difficult it is to shake off the picture of a machine that sucks in raw materials and spits out finished products. But actually, you are describing timeless relationships between numbers. Or that's what you seem to be saying.
Quoting TonesInDeepFreeze
I don't really understand this. If the lamp is neither off nor on at 12:00 (and still exists) then it must be in a third state of some kind. Or do you mean that it is not defined as on or off, which leaves the possibility that it must be in one state or the other, we just don't know which.
Quoting Deleted user
I don't get the difference. If mathematics applies to the physical world, surely it is true of it?
Quoting Deleted user
Yes. Different geometries apply in different contexts. That's only a problem if you think that just one of them must be absolutely true, which appears to be false.
By the premises, there is no third state. Indeed, even if not a premise but a definition:
Df. 'On' means 'not Off'
there is no third state.
Quoting Ludwig V
No, Thomson's argument is: The premises entail that at 12:00 the lamp is neither Off nor On, but the premises also include the stipulation that at all times the lamp is either Off or On, so the premises are inconsistent.
Quoting Ludwig V
No, it's not a matter of knowledge. Rather, at 12:00 the lamp is neither Off nor On, which contradicts that at all times the lamp is either Off or On.
Who says anything about probability when merely mentioning that .9... = 1.
We prove that .9... = 1, from the definition of the notation '.9...'.
'.9...' stands for the limit of a certain sequence, and that limit is 1.
Anyone is free to regard '.9...' with a different definition and to get different results accordingly. But in context of the ordinary mathematical definition, we prove that .9... = 1.
The adjective 'is potentially infinite' has no mathematical definition that I know of, including in alternative theories.
The adjective 'is infinite' is defined in mathematics.
The adjective 'is actually infinite' has no mathematical definition that I know of, including in alternative theories, unless it means simply 'is infinite'.
'is potentially infinite' is a notion about mathematics.
'is actually infinite', if not meaning simply 'is infinite', is a notion about mathematics.
Relax! I was talking about the traditional Aristotelian approach to infinity which was orthodox before Descartes but not since, so far as I know. Though I have since seen someone apparently still using the terms in Two Philosophers on a beach with Viking Dogs
Quoting TonesInDeepFreeze
Yes, I didn't think of the possible application of that idea to this discussion. I've only ever encountered it in the context of probability.
Quoting TonesInDeepFreeze
That's interesting. Can you refer me to a source?
Quoting TonesInDeepFreeze
I'm sorry. It's probably not worth pursuing, but I was struck by the point that "at all times the lamp is either Off or On" appears to be true while "the lamp is neither Off nor On" appears to be false, by reason of a failed referent. It's true by definition that a lamp is either off or on, so if some object is capable of being neither off nor on is not a lamp. The story is incoherent from the start. We cannot even imagine it.
This issue was actually resolved a long time ago by Aristotle, in his discussions on the nature of "becoming". What he demonstrated is that between two opposing states (on and off in this case), there is a process of change, known as becoming. This process is the means by which the one property is replaced by the opposing property. If we posit a third state between the two states, as the process of change, then there will now be a process of change between the first and the third, and between the third and the second. We'd now have five distinct states, and the need to posit more states in between, to account for the process of change which occurs between each of the five. This produces an infinite regress.
So what Aristotle proposed is that becoming, as the activity which results in a changed state, is categorically different from, and incompatible with states of being. Further, he posited "matter" as the potential for change. "Potential" refers to that which neither is nor is not. As what may or may not be, "potential" violates the law of excluded middle. So in the example, when the lamp is neither on nor off, rather than think that there must be a third state which violates the excluded middle law, we can say that it is neither on nor off, being understood as potential. This is the way that I understand Aristotle to have proposed that we deal with such activity, which appears to be unintelligible by violation of the law of excluded middle, neither having nor not having a specified property. The unintelligibility is due to a thing's matter or potential.
I won't refer you to a source.
I'll refer you to this:
Definition: .999... = lim(k = 1 to inf) SUM(j = 1 to k) 9/(10^j).
Let f(k) = SUM(j = 1 to k) 9/(10^j).
Show that lim(k = 1 to inf) f(k) = 1.
That is, show that, for all e > 0, there exists n such that, for all k > n, |f(k) - 1| < e.
First, by induction on k, we show that, for all k, 1 - f(k) = 1/(10^k).
Base step: If k = 1, then 1 - f(k) = 1/10 = 1(10^k).
Inductive hypothesis: 1 - f(k) = 1/(10^k).
Show that 1 - f(k+1) = 1/(10^(k+1)).
1 - f(k+1) = 1 - (f(k) + 9/(10^(k+1)) = 1 - f(k) - 9/(10^(k+1)).
By the inductive hypothesis, 1 - f(k) - 9/(10^(k+1)) = 1/(10^k) - 9/(10^(k+1)).
Since 1/(10^k) - 9/(10^(k+1)) = 1/(10^(k+1)), we have 1 - f(k+1) = 1/(10^(k+1)).
So by induction, for all k, 1 - f(k) = 1/(10^k).
Let e > 0. Then there exists n such that, 1/(10^n) < e.
For all k > n, 1/(10^k) < 1/(10^n).
So, |1 - f(k)| = 1 - f(k) = 1/(10^k) < 1/(10^n) < e.
(I saw an argument in a video that is much simpler, but I didn't get around to fully checking out whether it's rigorous. But arguments that subtract infinite rows are handwaving since subtraction with infinite rows is not defined.)
I was quite relaxed when I provided the information.
Quoting Ludwig V
The argument shows that the premises entail a contradiction, so at least one of the premises must be rejected. You can go back to the argument to witness it step by step. Best is to read Thomson's paper that is not long and not abstruse, free to download.
I agree with everything you wrote in this post.
Two presentations that are equivalent.
I would like to know how C2 and C3 are derived in Michael's version. That is rC1 and rC2 in my version.
But we get them anyway from my premise rP6 (the antecedent of Michael's C6). I'll show that presentation too (PRESENTATION2). And I think it is closer to Thomson's argument.
MICHAEL'S PRESENTATION
P1. Nothing happens to the lamp except what is caused to happen to it by pushing the button
P2. If the lamp is off and the button is pushed then the lamp is turned on
P3. If the lamp is on and the button is pushed then the lamp is turned off
P4. The lamp is off at 10:00
From these we can then deduce:
C1. The lamp is either on or off at all tn >= 10:00
C2. The lamp is on at some tn > 10:00 iff the button was pushed at some ti > 10:00 and <= tn to turn it on and not then pushed at some tj > ti and <= tn to turn it off
C3. If the lamp is on at some tn > 10:00 then the lamp is off at some tm > tn iff the button was pushed at some ti > tn and <= tm to turn it off and not then pushed at some tj > ti and <= tm to turn it on
From these we can then deduce:
C4. If the button is only ever pushed at 11:00 then the lamp is on at 12:00
C5. If the button is only ever pushed at 11:00 and 11:30 then the lamp is off at 12:00
C6. If the button is only ever pushed at 11:00, 11:30, 11:45, and so on ad infinitum, then the lamp is neither on nor off at 12:00 [contradiction]
TONESINDEEPFREEZE'S PRESENTATION:
Premises:
rP1: At all times, the lamp is either Off or On and not both.
rP2: The lamp does not change from Off to On, or from On to Off, except by pushing the button.
rP3: If the lamp is Off and then the button is pushed, then the lamp turns On.
rP4. If the lamp is On and then the button is pushed, then the lamp turns Off.
rP5: The lamp is Off at 10:00.
Conclusions:
rC1: If the lamp is On at a time T2 after 10:00, then it was pushed On at some time T1 that is both after 10:00 and before or at T2, and not pushed at any time that is both after T1 and before or at T2.
rC2: If the lamp is On at a time T1 after 10:00 then Off at a time T3 after T1, then it was pushed Off at some time T2 both after T1 and before or at T3, and not pushed at any time that is both after T2 and before or at T3.
Premise:
rP6: At 11:00 the button is pushed to turn the lamp On, at 11:30 Off, at 11:45 On, and alternating in that way ad infinitum.
Conclusion:
rC3: The lamp is neither Off nor On at 12:00. Contradicts rP1.
Again, I don't know how we derive Michael's C2 and C3 (my rC1 and rC2). But we don't need them anyway:
TONESINDEEPFREEZE'S PRESENTATION 2:
Premises:
rP1: At all times, the lamp is either Off or On and not both.
rP2: The lamp does not change from Off to On, or from On to Off, except by pushing the button.
rP3: If the lamp is Off and then the button is pushed, then the lamp turns On.
rP4. If the lamp is On and then the button is pushed, then the lamp turns Off.
rP5: The lamp is Off at 10:00.
rP6: At 11:00 the button is pushed to turn the lamp On, at 11:30 Off, at 11:45 On, and alternating in that way ad infinitum.
Conclusions:
rC1: If the lamp is On at a time T2 after 10:00, then it was pushed On at some time T1 that is both after 10:00 and before or at T2, and not pushed at any time that is both after T1 and before or at T2.
rC2: If the lamp is On at a time T1 after 10:00 then Off at a time T3 after T1, then it was pushed Off at some time T2 both after T1 and before or at T3, and not pushed at any time that is both after T2 and before or at T3.
rC1: The lamp is neither Off nor On at 12:00. Contradicts rP1.
So, we don't have to be concerned whether rP1-rP5 entail rC1-rC3. Rather, we see easily that rP1-rP6 entail rC1-rC3. It's a clean and correct inference that way.
So, unless we do have a proof of Michael's C2 and C3 from his P1-P4, he has his argument out of order: we need my rP6 in the premises. And that seems to be flow of Thomson's argument too.
I've bowed out of the supertask discussion, having not typed anything new in weeks. It would be inappropriate for me to comment on anything @Michael said, since he'd then be obliged to reply and we'd be right back in it again. @TonesInDeepFreeze merely made the point that 1/2, 1/4, etc only requires the rational numbers. Perfectly sensible observation.
Quoting Metaphysician Undercover
I'm fully supertasked out.
Quoting Metaphysician Undercover
Let's keep the Infinity theory in that thread. Well I'm not a moderator here so nevermind, do what you like. I prefer not to engage in these thread-hijacking points here. Every discipline has its terms of art, which confuse non-practioners. When a doctor tells you your liver is "unremarkable," that's great news and not an insult.
Oh I see. That's what I like about discussion forums. You can pick up a topic weeks or even months later.
Quoting Ludwig V
Oh my. We must have a conversation about probability sometime. You're wrong about that. 1 is a perfectly sensible probability. But worse, probability 1 events may be false. For example if you randomly pick a real number in the unit interval, it will be irrational with probability 1, even though there are infinitely many rationals.
Quoting Ludwig V
1 is a perfectly sensible probability. Your friend is misinformed. As Mark Twain said, if you don't read the newspapers, you're uninformed. If you do read the newspapers, you're misinformed.
Quoting Ludwig V
It's mathematically unhelpful to think of a infinite sequence as the "completion of an infinite number of tasks." It leads to confusion. It's not how mathematicians think about sequences.
Quoting Ludwig V
No, it's literally true. Of course math as a human activity is historically contingent. But math itself speaks to truths that are outside of time.
Quoting Ludwig V
Ok, but IMO it's deeper than a semantic point.
Quoting Ludwig V
Right. We don't even have good words to talk about things outside of time. Timeless present is a pretty good phrase.
Quoting Ludwig V
As in Wigner's famous paper on the "unreasonable effectiveness" of math in the physical sciences. If math doesn't actually refer to anything, why's it so useful?
Quoting Ludwig V
I have no idea. I have myself argued from time to time that 5 was not a prime number before there were intelligent beings to observe that fact. I don't actually believe that, but I've argued it.
Quoting Ludwig V
This was in reference to 0, 1, 2, ... existing "all at once" in PA. What ways are there to describe this? Is it a timeless present? That's a great locution.
Quoting Ludwig V
Yes. "Approach" is a term of art in mathematics. It has a specific technical meaning that is unambiguous. It is not the same as the everyday meaning.
Quoting Ludwig V
Term of art. A lovely legal phrase. Lawyers commonly have to deal with the jargon of whatever discipline a a particular dispute is about.
Quoting Ludwig V
There's a class math majors take called Real Analysis, where they teach you all this; and after which you are forever clear in your mind about things that were formerly vague and fuzzy. Sadly nobody but math majors takes this class, leading to so much confusion.
Quoting Ludwig V
Yes, that's a "function machine," a visualization when we teach functions to high schoolers. And of course mathematical functions are routinely applied to real world processes. A vending machine is a function of two variables: put in money and push a particular button, and the appropriate product comes out.
So the picture, or visualization of a function as a process or a machine is perfectly valid. The mathematical abstraction that strips away the process or machine interpretation is for the purpose of clarifying our ideas.
Quoting Ludwig V
Yes. The elements of a sequence have a timeless relation to the index set 1, 2, 3, ...
I haven't yet read all of Benacerraf's paper, but at least where he disscusses Aladdin and Bernard, it seems to me that he's not addressing Thomson's problem but only offering a different problem that does have an easy solution.
With Thomson's problem we have:
If the lamp is On at a time T2 after 10:00, then it was pushed On at some time T1 that is both after 10:00 and before or at T2, and not pushed at any time that is both after T1 and before or at T2.
and
If the lamp is On at a time T1 after 10:00 then Off at a time T3 after T1, then it was pushed Off at some time T2 both after T1 and before or at T3, and not pushed at any time that is both after T2 and before or at T3.
It seems to me that Benacerraf is skipping that condition. And so is the Cinderella example, which, if I'm not mistaken is a rewording of Benacerraf.
Next would be to examine whether your inference is correct that the problem shows that time is not infinitely divisible (or that it is not possible that time is infinitely divisible - and the modality there may make this more complicated). If I understand correctly, Thomson does't announce such a view about time, though, of course, what Thomson may believe doesn't determine our own conclusions.
@Michael's point, about which he and I disagree.
Quoting TonesInDeepFreeze
Don't believe so. But by expressing disagreement I invite rebuttal. I am supertasked out, really. Hope I have the strength to not get sucked in again.
If Benacerraf is not skipping the condition, then where does he recognize it? [EDIT: Actually he does address it, but, as far as I can tell, he gets it wrong when he addresses it.]
What essential difference is there between Aladdin/Bernard and Cinderella?
"A. Aladdin starts at to and performs the super-task in question just as
Thomson does. Let t1 be the first inistant 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 to and performs the super-task in question (on an-
other 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"
But that contradicts:
If the lamp is On at a time T2 after 10:00, then it was pushed On at some time T1 that is both after 10:00 and before or at T2, and not pushed at any time that is both after T1 and before or at T2.
and
If the lamp is On at a time T1 after 10:00 then Off at a time T3 after T1, then it was pushed Off at some time T2 both after T1 and before or at T3, and not pushed at any time that is both after T2 and before or at T3.
Benacerraf is saying the lamp gets switched in a way that is not possible given Thomson's conditions.
When we describe the events, we have to look closely and exactly at whether they may occur given Thomson's premises. We can't dissolve Thomson's argument merely by ignoring the premises of the argument.
/
Benacerraf:
"According to Thonmson, Aladdin's lamp cannot be on at t,
because Aladdin turned it off after each time he turned it on.
But this is true only of instants before tl!"
There he does recognize the premises, but, it seems to me, he mistakes them. The premises don't cover just what happens before 12:00. The premises state conditions that obtain at all moments whatsoever. The fact that certain conditions are specified for before 12:00 doesn't entail that all the rest of the conditions don't obtain at all times.
"Nothing whatever has been said about the lamp at t1
or later."
That seems to me to be incorrect. The premises state conditions that obtain at all moments whatsoever. The fact that certain conditions are specified for before 12:00 doesn't entail that all the rest of the conditions don't obtain at all times.
Benacerraf:
"The explanation
is quite simply that Thomson's instructions do not cover the state
of the lamp at t1, although they do tell us what will be its state at
every instant between to and t1"
The instructions don't need to specify what happens at 12:00. The instructions specify what happens at all moments and also what happens before 12:00, but what happens at 12:00 still must conform to the instructions that apply to all moments.
The issue is not that the instructions don't specify what happens at 12:00. The issue is that the instructions entail that at 12:00 the lamp is Off and at 12:00 the lamp is On. Thus the instructions are contradictory.
Please accept my regrets for not engaging. I have little interest in supertask puzzles in general, and this thread has long since exhausted any points I could possibly make.
The simple reasoning is that if time is infinitely divisible then pushing a button an infinite number of times within two minutes is theoretically possible. Pushing a button an infinite number of times within two minutes entails a contradiction and so isn't theoretically possible. Therefore, time is not infinitely divisible.
Although I think perhaps this variation of Zeno's paradox might be better at questioning the infinite divisibility of spacetime.
I'm not deeply versed in Aristotle, but my impression is that he did indeed resolve the issue, as it was understood in his time (and what more than that could he possibly resolve?). In doing so, he invented or discovered or recognized the concept of categories, which was a titanic moment in philosophy. It's a pity that there seem to be so many people around who are completely unaware of it.
Quoting Metaphysician Undercover
I think it would be more accurate to say "The apparent unintelligibility is due to a thing's matter or potential."
Quoting Metaphysician Undercover
I don't think that's quite right. It is true that if the lamp is on, it has the potential to be off, and if the lamp is off, it has the potential to be on. But that's not the same as having the potential to be neither off nor on. A lamp, by definition, is something that is on or off, but not neither and not both. There are things that are neither off nor on, but they are not lamps and the point about them is that "off" and "on" are not defined for them. Tables, Trees, Rainbows etc.
Quoting Metaphysician Undercover
I don't think that's quite right. The LEM does not apply, or cannot be applied in the same way to possibilities and probabilities. "may" does not usually exclude "may not". On the contrary, it is essential to the meaning that both are (normally) possible - but not both at the same time.
i.e. the lamp can't turn into a pumpkin.
i.e. the lamp is on if and only if the button is pushed (when the lamp is off) to turn it on (and not then pushed to turn it off).
Infinite divisibility doesn't entail a contradiction. Rather, infinitely divisibility along with the other premises entails a contradiction. Moreover, you are adding another premise (call it 'DT'): if infinite divisibility, then tasks can be performed at each of the infinitely many times. Therefore, we are entitled to question any of the premises, including the new one DT, not just infinite divisibility.
I'm not rejecting anything.
I'm saying:
(1) What is the proof of C2 and C3 from the premises? (Though we don't need it, if we adopt my rP6.)
(2) Instead of rejecting infinite divisibility, we may reject other premises instead.
That comment was directed at fishfry who claims that the lamp can turn into a pumpkin or spontaneously and without cause be on at 12:00.
Got it.
I'm deeply flattered. But that is far too much for me to grasp in less than a month or two.
Quoting TonesInDeepFreeze
Perhaps it would serve our purposes. I could probably get the point even if it isn't completely rigorous.
But let me explain why I need convincing.
In my book 0.9 + 0.1 = 1 and 1 - 0.1 = 0.9 and so 0.9 does not equal 1. There's a similar argument for 0.99 and 1 and so on. So for each element of 0.99999....., I have an argument that it does not equal 1. However, I see that your proof involves limits and I know that in that context words change their meanings. So I'm curious.
Quoting TonesInDeepFreeze
Well, it seems clear that at any specific time, it will be on or off depending on whether the button has been pushed an even number of times or an odd number of times since 11:00.
So at each of the times specified in the sequence, it will be on or off depending whether the number of times it has been pushed since 11:00 is odd or even.
Quoting TonesInDeepFreeze
The contradiction is created here - specifically in the last two words, which make it impossible to know whether it has been pushed an even or odd number of times since 11:00.
Which one do you think should be rejected?
If I said anything about that, I would be way out of my depth. So I'm afraid I shall have to ignore it - until another time, maybe.
Quoting fishfry
.. in the context of probability theory, that may be so. But I'm interested in probability in the context of truth and falsity, which is a different context. So when you say that 1 is a perfectly sensible probability, are you saying that probability = 1 means that the relevant statement is true? (I don't want to disappear down the rabbit hole, so I just want to know what you think; I have no intention of arguing about it.
I think this is a misunderstanding of the problem.
Say we accept that Thomson's lamp entails a contradiction; the lamp can neither be on nor off at 12:00.
I take this as proof that having pushed a button an infinite number of times is metaphysically impossible.
You seem to take this as proof that having pushed a button an infinite number of times is metaphysically impossible only if the premises are true.
As an example, let's say that our button is broken; pushing it never turns the lamp on. In such a scenario we can unproblematically say that the lamp is off at 12:00. But this does not then entail that it is possible to have pushed the button an infinite number of times.
We can imagine a lamp with two buttons; one that turns it on and off and one that does nothing. Whenever it's possible to push one it's also possible to push the other, and so if it's possible to have pushed the broken button an infinite number of times then it's possible to have pushed the working button an infinite number of times. Given that the latter is false, the former is also false.
Having pushed a button an infinite number of times is an inherent contradiction, unrelated to what pushing the button does. Having the button turn a lamp on and off, and the lamp therefore being neither on nor off at the end, is only a way to demonstrate the contradiction; it isn't the reason for the contradiction.
Which is also why Benacerraf's response to the problem misses the mark.
The pseudocode I provided a month ago helps explain this:
isLampOn is only ever set to true or false (and never unset) but the echo isLampOn line can neither output true nor false. This demonstrates the incoherency in claiming that while (true) { ... } can complete.
Changing echo isLampOn to echo true does not retroactively make it possible for while (true) { ... } to complete.
Having pushButton() do nothing does not make it possible for while (true) { ... } to complete.
It is metaphysically impossible for while (true) { ... } to complete, regardless of what happens before, within, or after, i.e. neither of these can complete:
Code 1
Code 2
Here's how I look at it. I think that everyone will agree that a formula is not about anything specific and, in itself is neither true nor false. x + y = z doesn't make any assertions, until you substitute values for the variables. So 2 +1 = 4 is false, but 2 + 3 = 5 is true. So there's a temptation to think it must be true of something. Hence realism. But 2 + 3 = 5 is itself like a formula in that once we specify what is being counted, it does make an assertion about the world - 2 apples + 2 apples = 4 apples. It is true of the world. Of course, 2 drops of water plus 2 drops of water doesn't make 4 drops of water, (until we learn to measure the volume of water). The domain of applicability and truth is limited.
Ok forget that. But 0 and 1 are perfectly legitimate probabilities. After all if I roll a die, the probability is 1 that it will be either 1, 2, 3, 4, 5, or 6, Right? Nothing degenerate or unusual about that. And the probability is 0 that it will show 7.
Quoting Ludwig V
Are you talking about credence, perhaps?
Quoting Ludwig V
No. Only that the event is certain, in the finite case; or "almost certain" to happen, in the infinite case.
Can you give me an example of what you mean? What kind of statements are you applying probability to?
Quoting Ludwig V
Never mind probability. You just startled me by denying the legitimacy or sensibility of 0 and 1 as probabilities.
I don't quite get what you mean here. Let's say there's something about reality which appears to be unintelligible. If we assign a name to that aspect, aren't we saying that there is actually something there which is unintelligible, and we've named it. This is to take a step further than simply that it appears as unintelligible.
Quoting Ludwig V
The point is that potential defies the laws of logic. That's why modal logic gets so complex, it's an attempt to bring that which defies the laws of logic into a logical structure.
The point I made, derived from Aristotle, is that whenever the lamp switches from on to off, or vise versa, there is necessarily a period of time during which it is changing (becoming). In other words, it is impossible that the switch from one to the other is instantaneous, and this is proven logically. In this 'mean time', the lamp is neither on nor off, and this defies the law of excluded middle. Dialethists would hold that it is both on and off, defying the law of noncontradiction.
Aristotle uses the concept of "potential" to explain his choice for defying the law of excluded middle rather than defying the law of noncontradiction. For him, the concept of "potential" is required to explain how something changes form having x property (being on), to not having x property (being off). "Potential" is a requirement of such a change, the thing cannot change without having the potential for change. However, this is a temporal concept, and the conclusion is that actualization requires a duration of time. So there is always a period of time between having x property (being on), and not having x property (being off).
What the lamp problem does not take into account, is that period of time between being on and off, during which it is changing. Assuming that the amount of time required to change from on to off, and vise versa, remains constant, then as the amount of time that the lamp is on and off for, gets smaller and smaller, the proportion of the time which it is neither, gets larger and larger. So at the beginning, when the time on and off are relatively long periods, the time of neither seems completely insignificant. But as the off/on actualization rapidly increases, the time of being on and off soon becomes insignificant in comparison to the time of being neither. The time of neither approaches all the time
Quoting Ludwig V
You only say this, because the time of change in which the lamp is neither on nor off is so short and insignificant that it appears to be nil. Aristotle demonstrated logically that it cannot be nil. So when we say things like "lamps are a type of thing which must be on or off, and cannot be neither", this is a statement about how things appear to be, and this facilitates much of our talk about such things. But when we get down to the way that things actually are, the way that logic tells us they must be, we can see that this way of allowing appearances to guide our speaking is actually misleading.
Quoting Ludwig V
I don't understand this. If a thing neither has nor has not the specified property, the excluded middle principle is violated (unless it's an inapplicable category). Potential itself neither is nor is not, and that's why we say it refers to what may or may not be. So "may or may not be" refers to the property we judge as in potential, and this says it neither is nor is not attributable to the thing.
I don't proffer an opinion on that. But I can see that presumably the most likely candidate is "At 11:00 the button is pushed to turn the lamp On, at 11:30 Off, at 11:45 On, and alternating in that way ad infinitum." At least intuitively it is the ripest and lowest hanging fruit. Or put pejoratively, at least intuitively it is the sore thumb.
But logically we may reject any of them. None of them are logical truths (though, "At all times, the lamp is either Off or On and not both" would be logically true as a conclusion from defining 'On' as 'not Off'.
(1) If a set of premises G entails a contradiction, and for any member P of G we have that G\{P} does not entail a contradiction, then we are logically free to reject any member of G. None of the premises are logically true (except "Either On or Off and not both" as conclusion from a definition "'On' means 'not Off'"), so we can reject any of them. For example, we could reject "The lamp does not change from Off to On, or from On to Off, except by pushing the button."
(2) We still don't have a satisfactory definition here of 'metaphysically impossible'.
(3) For what it's worth, if I'm not mistaken, Thomson does not conclude the time is not infinitely divisible, but rather he weighs in against the notion of super-tasks.
As a coda to Thomson's argument, you rely on the premise "If time is infinitely divisible then the super-task is possible.' [not a quote of yours]
But we may reject that premise.
It seems to me that "the lamp super-task is executed" entails "time is infinitely divisible". But the converse - "time is infinitely divisible" entails "the lamp super-task is executed" - at least requires an argument.
And, it seems to me, that analysis is even more difficult because it involves modalities. First, we have to distinguish between "time is infinitely divisible" and "it is possible that time is infinitely divisible". Second, it's not really, "the lamp super-task is executed" but "it is possible that the lamp super-task is executed". The argument needs to checked whether the modal inferences are correct.
Quoting Michael
No, I'm not claiming that.
Quoting Michael
Of course.
Quoting Michael
Whatever the validity of that, I don't see the point of it here.
Quoting Michael
It's not a contradiction in and of itself. Rather, it is inconsistent with the other premises (especially that the infinite number of executions occurs in finite time).
Quoting Michael
You would need to tell me the difference between a demonstration of a contradiction and the reason for a contradiction. For example, if someone asks "What is the reason that the unrestricted comprehension schema with the separation schema yields a contradiction?" then my best response would be to show a demonstration.
Quoting Michael
As far as I can tell, it's off-base because it doesn't address the premises of the lamp.
And related arguments against Thomson are that it is not problematic that the premises don't provide for concluding whether the lamp is Off or On at 12:00. But that misses the point that it is not that it is problematic that the lamp's state is undetermined, but rather that Thomson's argument shows that the lamp is neither Off nor On. Not that it is undetermined what the state is, but rather that is determined that the state is neither Off nor On.
/
I don't know enough about coding to have a comment on your pseudocode.
/
A while back you gave the argument in quite succinct form:
"P1. The lamp is turned on and off only by pushing the button
P2. If the lamp is off and the button is pushed then the lamp is turned on
P3. If the lamp is on and the button is pushed then the lamp is turned off
P4. The lamp is off at t0
P5. The button is pushed at successively halved intervals of time between t0 and t1
P6. The lamp is either on or off at t1"
But you add to that argument two things:
(a) We must reject P5.
But each of the premises is required for the contradiction, so we can reject any one of them rather than P5. Granted, P5 does stand out as the candidate we would intuitively reject, but it is not logically required that it is the one we reject. For example, famously:
(U) ExAy yex
(S) ExAy(yex <-> ~yey)
yield a contradiction.
But to dispel the contradiction it is not logically required that we reject (U) to keep (S) when we could reject (S) to keep (U). We may have reasons for preferring that we reject (U) rather than (S), but that is not a demonstration that (U) is logically impossible or even that it is false.
(b) If time is infinitely divisible, then the super-task may be executed.
But that is not logically true either. It may be the case that time is infinitely divisible but still the super-task cannot be executed. Moreover, the modality "may be" slips in there, so the argument requires that it is made clear that the modal inference is permitted.
An infinite sequence of operations is by definition an endless sequence of operations. An endless sequence of operations does not come to an end. That's what makes the premise of a supertask an inherent contradiction.
Having the operation be to push a button, and having this button turn a lamp on and off, is simply a way to make this inherent contradiction even clearer.
If you accept that this proves that this button cannot have been pushed an infinite number of times then what is the reasoning behind the claim that if some wizard steps in at 12:00 to magically turn the lamp on then this retroactively makes it possible to have pushed this button an infinite number of times? Let's even assume for the sake of argument that this wizard will only appear with a probability of 0.5, and that this is determined only at exactly 12:00, i.e after the performance of the supertask. It must already be possible for the supertask to be performed for him to even appear, and so his appearance cannot retroactively make the supertask possible, even if half the time it resolves the secondary contradiction regarding the state of the lamp at 12:00.
I don't think you're really grasping what distinguishes a supertask from an abstract infinite sequence.
I didn't say they end.
Quoting Michael
Again, the contradiction comes from the conjunction of the premises. It is not a given that it is a contradiction in and of itself that infinitely tasks are executed in finite time. It's quite unintuitive that infinitely many tasks an be executed in finite time, but to show that doesn't entail that it is a contradiction in and of itself.
Quoting Michael
I never claimed any such thing. It's a straw man, even if unintentional.
What I said is that if we drop the premise that the lamp is only turned on by the button, then we don't get the contradiction. The point of that is that we are not logically obliged to reject only one certain premise. Note that I am not committing the fallacy of not addressing Thomson's premises. Rather, I am pointing out that rejecting one of the premises to avoid contradiction must then allow rejecting any other premise to avoid contradiction.
When I saw that you had put care into your numbered arguments, I surmised that you were interested in pursuing rigor. So in that regard, I'm examining all your reasoning.
Again (new numbering):
(1) Since none of the premises are logically true, but they yield a contradiction, we may reject any one of them to avoid contradiction. We are not logically bound to reject the one that happens to be least intuitive.
(2) Infinitely divisibility of time does not entail executability of denumerably many tasks in finite time, even though, executability of denumerably many tasks in finite time entails infinite divisibility of time.
(3) We don't have a satisfactory definition of 'metaphysical possibility' here.
(4) The argument is more complicated than appears with only a cursory look, since it involves the modality of 'possible'.
(5) If I'm not mistaken, Thomson does not conclude that time is not infinitely divisible. So, heuristically, we may wonder why that is if your conclusion actually follows as ineluctably as you claim.
(6) I wonder why you don't note my point about continuousness and density, which I mentioned to help sharpen your argument.
A supertask is an infinite sequence of operations that ends in finite time.
Quoting TonesInDeepFreeze
One of the contradictions does; the state of the lamp at 12:00. This isn't the only contradiction. The other contradiction is the inherent contradiction of an endless sequence of operations coming to an end. The former is simply a tool to better demonstrate the latter.
Finding some way to resolve the former does not also resolve the latter.
Just in case you missed my edit to my previous post:
Let's even assume for the sake of argument that this wizard will only appear with a probability of 0.5, and that this is determined only at exactly 12:00, i.e after the performance of the supertask. It must already be possible for the supertask to be performed for him to even appear, and so his appearance cannot retroactively make the supertask possible, even if half the time it resolves the secondary contradiction regarding the state of the lamp at 12:00.
Quoting TonesInDeepFreeze
See here.
All I mean by it is that supertasks are more than just physically impossible. No alternate physics can allow for them.
As a different example, consider the grandfather paradox. I don't just take this as a proof that one cannot travel back in time and kill one's grandfather before one's father is born; I take this as a proof that one cannot travel back in time.
The premise of having one kill one's grandfather before one's father is born is just a tool to prove the impossibility, not the reason for the impossibility.
You answered pretty fast. That's your prerogative. But it make me wonder whether you're giving much thought to my remarks, as still it would be your prerogative not to. So I'll take the same prerogative.
The definition of a super-task is as you say. But your listed premises don't say anything about completion or ending.
Quoting Michael
The contradiction is: The lamp is either On or Off T 12:00 and the lamp is neither On nor Off at 12:00.
But that contradiction comes from a set of premises, each of which is not logically true, and dropping any one of the premises blocks deriving the contradiction. It would help if you would at least tell me that you understand that.
Quoting Michael
(1) The premises don't say it comes to an end. It would help if you would at least tell me that you understand that.
(2) It is begging the question merely to declare it is a contradiction that denumerably many tasks can be executed in finite time. Indeed, the argument itself doesn't declare that it is a contradiction. Rather, the argument derives a contradiction from that premise along with other premises.
Again, the example I gave:
AxEy yex
ExAy(yex <-> ~yey)
entails a contradiction, but it doesn't entail that either of the above is itself a contradiction.
Even most minimally:
P
~P
entails a contradiction, but that doesn't entail that either P or ~P is a contradiction.
It would help if you would at least tell me that you understand this.
Quoting Michael
There's no just in case that you missed my own post. You missed that I said I don't argue in any such way that is knocked down as a straw man as you have.
/
The link doesn't go to a defininition. It merely says that metaphysical possibility may be logically possibility and that there's another notion that the article describes ostensively. So is it just the same as logical possibility, and if not what is a proper definition that is not merely ostensive?
Finding some way to resolve the former does not retroactively resolve the latter.
Quoting Michael
So, yes, clearly taking your prerogative to answer so quickly did result in your not even taking note of what I said, let alone taking a moment to understand it.
I am not "finding some way to resolve the former [to] retroactively resolve the latter." You can read my post again to see my explanation.
The infinite button pushes ends after two hours. That's the premise of Thomson's lamp (albeit minutes in his specific case). In his own words, "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?".
If, as per the premise, I only push the button at 11:00, 11:30, 11:45, and so on ad infinitum, then I am no longer pushing the button at any time after 12:00. My infinite button pushes has allegedly ended.
The very thing we're discussing is the possibility of supertasks, i.e. can an infinite sequence of operations end in finite time?
Quoting TonesInDeepFreeze
That's one of the contradictions. If one drops or adds or changes any premises, e.g. by stipulating that the lamp spontaneously and without cause turns into a pumpkin at 12:00, then you have resolved the contradiction regarding the state of the lamp at 12:00, but doing so does not then allow for the possibility of supertasks; it does not resolve the contradiction in claiming that an infinite sequence of button pushes has come to an end.
Quoting TonesInDeepFreeze
It's simply true by definition. An endless sequence of operations cannot end. An infinite sequence of operations is an endless sequence of operations. An infinite sequence of operations cannot end.
Quoting TonesInDeepFreeze
I'm not the authority on the matter. I am simply arguing that supertasks are more than just nomologically impossible. I use the phrase "metaphysical impossibility" rather than "logical impossibility" simply because it's the weaker claim. Call it hedging my bets if you will.
P2. If the lamp is off and the button is pushed then the lamp is turned on
P3. If the lamp is on and the button is pushed then the lamp is turned off
P4. The lamp is off at t0
P5. The button is pushed at successively halved intervals of time between t0 and t1
P6. The lamp is either on or off at t1
But actually, if we look at the mechanics of the inferences toward the contradiction, we see that we need that the lamp is never both on and off. So I would write.
P1. The lamp is turned on and off only by pushing the button
P2. If the lamp is off and the button is pushed then the lamp is turned on
P3. If the lamp is on and the button is pushed then the lamp is turned off
P4. The lamp is off at t0
P5. The button is pushed at successively halved intervals of time between t0 and t1
P6. At all times the lamp is either on or off and not both.
Then we derive a contradiction:
C1. The lamp is either on or off, and the lamp is neither on nor off.
Now, what premise do we delete to avoid the contradiction? Since none of them are logically true, and each is needed in the derivation of the contradiction, we may delete any one of them.
P5 is an inherent contradiction, just as travelling back in time is an inherent contradiction.
The lamp being neither on nor off at t[sub]1[/sub] and killing one's own grandfather before one's father is born are secondary contradictions to prove the inherent contradictions.
The possibility of P5 does not depend on whether or not P1-P4 are true, e.g. having the button be broken does not make it possible to push the button an infinite number of times within two hours.
You're stuck thinking I'm making a certain kind of argument, but I am not. You're not thinking about what I've specifically written, as probably you take me to be making a version of other arguments around. If you're not going to take my arguments as given, then there's no rational inquiry to be had.
But I'll address these two:
Quoting Michael
But completion is not in your premises.
Quoting Michael
That is begging the question. It is begging the question to rule that there can't be denumerably many tasks executed in finite time. You haven't proved: If there are denumerably many tasks executed in finite time then there is an end to their executions. So you have to either prove it or add it as a premise.
Quoting Michael
It's not much of a claim if it is not defined.
This is my argument.
Notice the antecedent of C6: "If the button is only ever pushed at 11:00, 11:30, 11:45, and so on ad infinitum...".
If I am only ever pushing the button at these times then I am not pushing the button at 12:00 or at any time after 12:00. Therefore, my (infinite/endless) button-pushing has ended by 12:00.
That's what supertasks are. They are an inherent contradiction, irrespective of what pushing the button actually does. It is as metaphysically impossible to have performed a supertask on a broken button as it is metaphysically impossible to have performed a supertask on a working button. Having the button turn the lamp on and off, like killing my grandfather before my father is born, is just a means to better demonstrate this impossibility and not the reason for it.
You haven't paid attention to my answer to that. Now, you're arguing by mere assertion and repeated mere assertion.
Moreover, if P5 is deemed in and of itself contradictory, then we don't need an argument with other premises to derive a contradiction. If P5 is, from the outset declared a contradiction, then we don't have to bother with the bloody lamp at all.
You're not thinking about my points as it seems you just lump them in with other people's arguments. You are skipping key points. You argue by question begging and mere assertion. And now you link me to a post from eleven days ago that I had already addressed in detail.
As to C6, you put the halving ad infinitum as an antecedent in a conclusion, which is fine. But it's equivalent to just making it a premise. And I mentioned that a while ago, and mentioned why it is more stark to make it a premise, but you ignored that too. Your terse argument that I've recently mentioned and your argument eleven days ago are essentially the same: It doesn't matter whether you put the halving ad infinitum as an antecedent in a conclusion or as a premise - it's logically the same.
Anyway, we're going in circles as you skip my arguments while reasserting your own. When I saw that initially you were taking care to make numbered arguments, I got interested and thought you might be open to more scrutiny. But I can see you're not, as you only keep repeating assertions and not actually thinking about the replies to them. It seems your primary interest is to persist that you are rigtht and not to truly think through objections. So, bye for now.
Yes, it makes no difference if it's an antecedent in a conclusion or as a premise. Either way, the supertask is the completion/end of an infinite/endless sequence within finite time (e.g. I have stopped pushing the button by 12:00) and is an inherent contradiction, irrespective of what the task is. Having the task be to push a button that turns a lamp on and off is just a means to demonstrate the impossibility of a supertask and not the reason for its impossibility, and neither having the button be broken nor having the lamp spontantously turn into a pumpkin allows for the supertask to be possible.
No, it doesn't. It's called 'the deduction theorem'. For example:
P, Q, R |- S
is equivalent with
P, Q |- R -> S
Huh? I'm reiterating/agreeing with your claim that "it doesn't matter whether you put the halving ad infinitum as an antecedent in a conclusion or as a premise - it's logically the same"?
I was rushing. My mistake.