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

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

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

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

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

What natural number did I not say? — Michael

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

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

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

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

The sequence has the well-known limit 0.

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

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

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

So main points;

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

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

Obviously the above is fallacious. — Michael

Fallacious! Non sequitur! Metaphysically impossible! Nonsense!

These are words. They are not arguments.

It is metaphysically impossible — Michael

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

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

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

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

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

I found that discussion very helpful. — Ludwig V

Glad to hear that.

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

Obviously the above is fallacious. It is metaphysically impossible to have recited the natural numbers in descending order. — Michael

I already agreed with this, because limit ordinals do not have immediate predecessors.

The fact that we can sum such an infinite series is irrelevant. And the same is true of your version of the argument. — Michael

If you would engage in your private time with the 60 second puzzle, you would see that each number is spoken at a specific, calculable time; that there is no first number that's not spoken; and therefore every number is spoken.

It's not productive for me to give a high-school level inductive argument and for you to say "nonsense" and "metaphysically impossible" without ever engaging with the argument.

Please read the Wiki page on mathematical induction and ask questions as necessary, and challenge yourself to engage with the argument.

Ask yourself: What is the first number not spoken? If you ask yourself that enough times, you may have an epiphany.

No you haven't. Your premise begs the question and simply asserts that all the natural numbers have been recited within 60 seconds. — Michael

Name the first one that's not. It's a trivial exercise to identify the exact time at which each natural number is spoken. "1" is spoken at 60, "2" at 90, "3" at 105, "4" at 112.5, and so forth.

Can you not see that we can calculate the exact time at which each number is spoken?

I did not "simply assert" all the numbers are spoken. I proved it logically. Induction works in the Peano axioms, I don't even need set theory.

If you work through this example you will obtain insight.

No, we're reciting the numbers in descending order. It's impossible to do, even in principle. The fact that we can assert that I recite the first number in N seconds and the second number in N/2 seconds and the third number in N/4 seconds, and so on ad infinitum, and the fact that the sum of this infinite series is 2N, doesn't then entail that the supertask is possible.

That we can sum such an infinite series is a red herring. — Michael

You're right that we can't "name all the numbers" going backward. but that's obvious. There's no largest number and limit ordinals don't have immediate predecessors.

It's pointless for you to snap back a minute later arguing with well established mathematical facts. I gave a solid inductive argument that by the premises of your 60 second puzzle, all the numbers will be spoken. That's because there's no first number that won't be spoken. If you doubt that, then name a number that's not spoken.

I ask you to read carefully what I'm writing, and think about it.

Did you ever learn mathematical induction? If yes, I gave a standard inductive argument. If no, that's a good starting point and I'll be happy to give a summary. I gave you the Wiki link. I can't argue well established facts with you.

In the puzzle you gave, every number must be spoken. In fact we could calculate, if we cared to, the exact time at which it's spoken.

Please give this some thought.

What number won't be spoken?

It begs the question. Your premise is necessarily false. Such a supertask is impossible, even in principle, to start. — Michael

Did you learn mathematical induction in school? Please review that. Please take the time to understand the argument I made.

Under the premises of the problem you posted, there is no number that does not get spoken.

It's imperative that you understand that. It's pointless for you to disagree. You must show that there's a number that did not get spoken. If you can't do that, then every number gets spoken.

You just listed five rational numbers and are claiming that this is proof of you reciting all the natural numbers in descending order? — Michael

I did not make that claim. I said I counted backward from a limit ordinal. That's easy. It's always a finite number of steps.

You're talking nonsense. — Michael

I'm counting backward from a limit ordinal. Very standard math.

What number do you recite after 1? — Michael

7/8 will do just fine. I necessarily had to jump over all but finitely members of the sequence.

Of course I can not count ALL the numbers backward. That's impossible. That's because limit ordinals do not have predecessors. That's the definition of a limit ordinal, an ordinal that does not have an immediate predecessor. So it's your challenge that's nonsense.

But please, I'm asking you to sit down and think about the inductive argument I made.

Counting forward with your 60 second idea, which number won't be spoken?

We can certainly say "1". And if we say n, we can say n + 1. This is high school mathematical induction. Please tell me you learned this. If not, that would explain your confusion. But if you made it through high school math (do they still teach induction in high school? What do I know) then you have the means to understand the argument.

Please take the time to think it through. What number can't be spoken?

ps -- I looked it up. Perhaps induction is not universally taught in high school, and it doesn't come up in calculus.

Do you know mathematical induction? It's a row of dominos.

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

Ok. See also ↪J 's other thread. — Banno

Thanks much.

Because it begs the question. — Michael

I go 1 at 60, 2 at 30, etc.

Name the first number that I fail to count

Third time I'm asking you the question. (At ever decreasing intervals of time!)

This is a standard inductive argument. If it's impossible to name the first natural number at which a property fails to hold, the property must hold for all natural numbers. Think back to when you learned inductive proofs in school. I can name 1. And if I name n, I can name n + 1. Therefore I can name all the numbers. Counterintuitive though it may be, it's true. You learned this in high school.

Please give this argument some thought.

That's not counting down from infinity. — Michael

You have no proof or evidence. On the contrary, the mathematics is clear.

I don't know what you mean that supertasks are nonterminating by definition.
— fishfry

Tasks are performed ad infinitum. I never stop counting. There's always another number to count. — Michael

Did I not move you, surprise you, convince you, that if you count 1, 2, 3, ... successively halving the time intervals, that you will indeed count every single natural number in finite time? If not, why not?

I am still waiting for you to name the first number I didn't count.

This is a standard inductive argument. To prove that a property holds for all natural numbers, I show the impossibility of there being a first number where the property fails.

I'm talking about reciting the numbers. So imagine someone reciting the natural numbers up to infinity. Now imagine that process in reverse. That's what I mean by someone counting down from infinity. — Michael

But counting backward from infinity is always finite! I showed you how that works, counting backward from 1 in the ordered set <1/2, 3/4, 7/8, ..., 1>

In fact this is true of all the transfinite ordinals. It's only finitely many steps backward from any transfinite ordinal, no matter how large. That's because stepping back from any limit ordinal (defined as an ordinal without an immediate predecessor) necessarily jumps over all but finitely many elements of the sequence that led up to it.

It is a non sequitur to argue that because we can sum an infinite series with terms that match the proposed time intervals that it is possible to have counted down from infinity. It is impossible, even in principle, to start such a count. The maths doesn't change this. — Michael

It's easy, I'll count backward from infinity right here on a public Internet forum, in plain view of the world.

1, 15/16, 7/8, 3/4, 1/2. Done. My first step necessarily jumped over all but finitely many elements of the infinite sequence. It must be that way.

That's because the first step backward from any limit ordinal necessarily jumps over all but finitely members of the sequence whose limit it is.

Counting backward from infinity is easy, and always finite!

Davidson is just the ubiquitous On the very idea of a conceptual scheme. — Banno

Not ubiquitous to me, I'm a philosophical dummy. I'll Google around.

There's a prima facie disagreement here, but I think it is on the surface only, that Midgley is espousing something not too dissimilar to Davidson's anomalism of the mental. — Banno

I'm way out of my depth. I will do some surfing and maybe glean some clues.

A quick Google search yielded:

What is Davidson's summary of the very idea of a conceptual scheme?

Davidson attacks the intelligibility of conceptual relativism, i.e. of truth relative to a conceptual scheme. He defines the notion of a conceptual scheme as something ordering, organizing, and rendering intelligible empirical content, and calls the position that employs both notions scheme‐content dualism.

and my eyes glazed over. I'll check out the thread you linked. Thanks for the pointers.

Well ok, then why don't I complete a supertask when I walk across the room, first going halfway, etc.? Can you distinguish this case from your definition?
— fishfry

If supertasks are impossible and motion is possible then motion isn't a supertask. — Michael

I don't find that satisfactory. It only casts doubt on the premise "if supertasks are impossible."

I agree with you that the lamp and staircase and other related puzzles are qualitatively different than Zeno's paradoxes of motion, so perhaps in that sense you want to reserve the word supertask for the former. But your definition is "completing a countably infinite number of tasks in finite time," and walking across the room seems to satisfy that definition.

Not to mention that, if we take the real numbers as a model of space, we pass through uncountably many points in finite time. That's another mystery.

* You have not convinced me or even made me understand your reasoning that supertasks are "metaphysically impossible" or that they entail a logical contradiction.
— fishfry

By definition supertasks are non-terminating processes, therefore you've gone wrong somewhere if you conclude that they can terminate after 2N seconds. — Michael

I don't know what you mean that supertasks are nonterminating by definition. Just thinking mathematically for a moment, limits "terminate" a sequence in the sense that 1 is the terminus of the sequence 1/2, 3/4, 7/8, ... The limit 1 is not part of the sequence, but we can imagine the 1 stuck at the end of an ordered set, as I have been doing, and it's perfectly sensible.

In other words supertasks are nonterminating, but they definitely may have a terminal state; just as a convergent mathematical sequence has no final term, yet has a limit. Is my analogy unsatisfactory with respect to your conception of supertasks?

Also I think the clearest example I gave was that of having counted down from infinity. We can assert (explaining what happened in reverse) that I recited 0 after 60 seconds, recited 1 after 30 seconds, recited 2 after 15 seconds, recited 3 after 7.5 seconds, etc., and we can say that we can sum an infinite series with terms that match the described (and implied) time intervals, but it doesn't then follow that we can have counted down from infinity; we can't even start such a count. — Michael

I don't follow how you are counting down from infinity. In fact when you count down from infinity, it's always only finitely many steps back. If I take the ordered set <1/2, 3/4, 7/8, ..., 1> and I start at 1, my first step backwards jumps over all but finitely many elements of the sequence, and it's always only finitely many steps back from 1 to 1/2.

[Per my recent convo w/@No Axioms I am using angle brackets to denote ordered sets].

You did lose me when you said that counting 0, 1, 2, ... is "counting down from infinity." I did not understand that example when you gave it earlier. Mathematically, the ordered set <1, 2, 3, ...> exists, all at once. Its counting is completed the moment it's invoked into existence by the axiom of infinity.

But let me ask you this. Suppose I say 0 at 60 seconds, and 1 at 30 seconds, and 3 at 15 seconds, and so forth.

Now I claim that after 120 seconds (the sum of the series) I have counted all the natural numbers!

Yes I claim that. And as proof, I challenge you to name the first number that I did not count.

Since you can not do that, I have indeed counted all the natural numbers.

The mathematics is evidently a non sequitur — Michael

I either don't understand what you mean, or I strenuously disagree.

Explain please?

, and it's a non sequitur in the case of having counted up to infinity as well. — Michael

I just proved to you, using a very standard inductive argument, that I can indeed count all the natural numbers as you described, in intervals of 60, 30, etc. Because you can not name any number I can't count. Did I count 47? Sure. Googolplex? Sure. Graham's number? Sure. There is no number that I didn't eventually count. Therefore I counted them all.

It brings out the conflict in my own arguments, between Midgley and Davidson, and provides something of a logical frame for that discussion. No small topic. — Banno

Have you got a reference to Midgley and Davidson? Is there an interesting professional discussion of these issues?

I said I had no problem with any of that.
Is it a belief thing, like it is some kind of religious proposition or something? "Hey, I'm going rogue here and will suspend belief that 7 is a factor of 35". — noAxioms

I'm making the point that you are perfectly willing to accept {1/2, 3/4, 7/8, ..., 1} as a valid set that contains an infinite sequence and its limit; but you are having trouble accepting {1, 2, 3, ..., $\omega$}, simply because it's far less familiar. But in terms of their order, they are exactly the same set. They have very different metric properties; but strictly with respect to order, they are two different representations of the same ordered set.

Treating infinity as a number, something you didn't do in your unionized set above — noAxioms

Transfinite ordinal numbers are numbers. It's just a matter of expanding one's concept of a number. $\omega$ is a number. It's the first transfinite ordinal number. I am casually calling it a "point at infinity," but if that bothers you, just think of it as 1 in the set {1/2, 3/4, 7/8, ..., 1}. It's exactly the same thing wearing a different suit of clothes.

It's an infinite sequence. I stuck the number 1 on the end.
Yea, when it normally is depicted at the beginning. From what I know, a set is a set regardless of the ordering. There must be a different term (ordered set?) that distinguishes two identical sets ordered differently, sort of like {1, 3, 5, 7 --- --- 8, 6, 4, 2} — noAxioms

Yes, ordered set. I have been casually using the curly braces, but you are absolutely correct. {1/2, 3/4, 7/8, ..., 1} has no order, I could stick the 1 in the middle or at the beginning and it would be the same set, but I'd lose the order that I consider important.

Perhaps a notation like <1/2, 3/4, 7/8, ..., 1> would be better, to indicate an ordered set. You are absolutely right. I did not want to add any more complications earlier, but the curly braces are inaccurate in the way I'm using them. I'm speaking of ordered sets. So I'll use angle brackets from now on.

The entire set is ordered by the usual order on the rational numbers. So why is it troubling you that I called 1 the "infinitieth" member of the ordered set?
It violates thebijunction. You can't say what number comes just before it, which you can for any other element except of course the first. You can do that with any other element. — noAxioms

Well then here yet another representation of the same idea. Suppose I reorder the natural numbers

<1, 2, 3, 4, ...>

by putting 1 at the end, so that I have:

<2, 3, 4, ..., 1>

You can see that I still have a bijection. As you noted, sets don't have order, so it's still the same set.

Note that I no longer have an order-preserving bijection. I merely have a set bijection. I can still correspond 1 to 1, 2 to 2, and so forth. But I can't do it in an order-preserving manner.

But now I have another representation of an ordered set that consists of an infinite sequence followed by a "point at infinity," or a largest element. That largest element does not have a predecessor, you are right about that.

And in fact we have a name for that. In ordinal theory, an ordinal with a predecessor is a successor ordinal. And an ordinal without a predecessor is a limit ordinal. So your intuitions are spot on.

OK, but what problem does it solve? It doesn't solve Zeno's thing because there's no problem with it. It doesn't solve the lamp thing since it still provides no answer to it. — noAxioms

Ah yes, why am I doing all this?

It solves the lamp problem. The lamp state is a function on <1/2, 3/4, 7/8, ..., 1> defined as "on" at 1/2, "off" at 3/4, "on" at 7/8, and so forth.

But now we see (more clearly, IMO) that the state at 1 is simply undefined. The statement of the problem defines the lamp state at each element of the sequence; but does NOT define the state at the limit.

We also note that there is no way to make the sequence 0, 1, 0, 1, ... continuous.

And since you didn't tell me what is the state at 1, and there is no natural way to define the state at 1, I am free to define the state at 1 any way I like. And inspired by Cinderella, I define the state of the lamp at 1 to be a plate of spaghetti. That's the solution to the problem. The final state is anything you like. It doesn't even have to be on or off since it's not a real lamp, just as Cinderella's coach is not a real coach. The lamp problem is every bit as much a fairy tale as Cinderella.

So for many of these supertask problems, the ordered set <1/2, 3/4, 7/8, ..., 1> is the natural setting for the problem.

Note that the staircase is different. The walker is on step 1, on step 2, etc. So the natural, continuous way of completing the sequence is to say that the walker is at the bottom of the stairs. This is totally different than the lamp, which can not be made continuous or sensible in any way at all.

So my entire point is that <1/2, 3/4, 7/8, ..., 1> is the natural way to think about these problems. The question is always: how did we define the state at the elements of the infinite sequence; and then, how are we free to define the final state at the limit.

Nobody's asking the particle to meaningfully discuss (mathematically or not) the step. It only has to get from one side to the other, and it does. Your argument is similar to Michael wanting a person to recite the number of each step, a form of meaningful discussion. — noAxioms

I'm not sure what you mean by referring to the subjective state of the particles. When Newton wrote down his great law of gravitation, he did not care how the masses feel about it. I'm not following your analogy.

Maybe we live in a discrete grid of points -- which would actually resolve Zeno's paradoxes.
It would falsify the first premise. Continuous space falsifies the second premise. Zeno posits two mutually contradictory premises, which isn't a paradox, only a par of mutually contradictory premises,. — noAxioms

I confess to not knowing the answer to Zeno. It's a clever argument. Unless the answer is that we satisfy Zeno and execute a supertask every time we walk across the room. But @Michael objects to that, for reasons I don't yet understand.

But I can say "for all we know, ....", and then there's no claim. I'm not making the claim you state. I'm simply saying we don't know it's not true. I even put out my opinion that I don't think it's true, but the chessboard thing isn't the alternative. That's even worse. It is a direct violation of all the premises of relativity theory (none of which has been proved). — noAxioms

Some speculative physicists (at least one, I believe) think the world is a large finite grid. It's not out of the realm of possibility as I understand it. I think I read that in Penrose's The Road to Reality. And if Sir Roger thinks it's good enough to put in a book, it must be of interest.

In other words the chessboard universe is not ruled out by any known theory or experiment. And we know that quantum and relativity have yet to be integrated, and perhaps that's a clue.

A supertask is "a countably infinite sequence of operations that occur sequentially within a finite interval of time."
— Michael
Yea, I don't know how that could have been lost. I don't think anybody attempted to redefine it anywhere. — noAxioms

Yes ok but then ... how is walking across the room by first traversing 1/2, then half of the remaining half, etc., not a supertask? I don't understand this point.

I'm not quite sure what you mean by "believe in the rational numbers." — keystone

You confused me a while back. You said you don't believe in the real numbers [or some similar wording].

So I asked you, what are those symbols 0, .5, 1, and so forth? If they're not real numbers, what are they?

That's why I asked you if you believe in the rational numbers. If you do, then you have to also believe in the reals, since the reals are constructed from the rationals. If you don't, then again I ask you what are 0, .5, and 1?

From a top-down perspective, there's no need to assert the existence of either R or Q, especially since all the subsets within the enclosing 'set' are finite. — keystone

You have been freely using the symbols 0, .5, and 1. If they are not real, and they are not rational, then I don't know what those symbols mean. Can you define them?

"... all the subsets within the enclosing 'set' are finite"???? Means what? Lost me there.

If you suggest that this enclosing 'set' is infinite, then we must rethink our definition of what an 'enclosing set' actually is in this context. I was hoping to put this particular discussion aside for now, as it will likely divert attention from our main focus. — keystone

You're the one with some notion of enclosing set. A metric space is a set with a distance function. If it lives in a larger ambient set, then you have to say what that is. You started a long time ago saying something like "the metric doesn't apply outside the metric space." Ok that's true, but what is outside? You have to say what that is.

Regarding Dedekind cuts, they involve splitting the infinite set of rational numbers into two subsets. This presupposes both the existence of an infinite entity (Q) and the completion of an infinite process (the split). If one rejects the concept of actual infinity, then it's questionable whether real numbers necessarily follow from rational numbers. — keystone

Ok fine. You reject the real numbers. You already said that.

So I asked you, do you believe in the rational numbers. And you asked me what I mean by that!

If you use symbols like 0, .5, and 1, you have to say what they are.

So, do you believe in the rational numbers? Is that the number system we're working in?

In which case I have to echo @jgill's excellent question as to whether you accept intervals like [pi, pi + 1], and if not, why not.

However, the discussion about actual infinity and the nature of real numbers could go on endlessly. — keystone

You could bring it to a quick conclusion by saying, "Yes, we are working in the rational numbers."

But you won't even say that! Leaving me totally confused.

I acknowledge that these concepts are crucial for a bottom-up approach, but can we instead focus on seeing how far a top-down perspective—devoid of actual infinities and traditional real numbers—can lead us? In the top-down view, reals hold a special role, just not as conventional numbers. — keystone

Sure. Then what are these funny symbols 0, .5, and 1 that you keep on using? What do your interval notations denote?

If you're working in the rationals that's fine, but when I asked you about it, you asked me what I meant by the question.

You are the one who started at 0, then got to (0, .5), and then magically completed a limiting process to get to .5. I ask again, how is that accomplished?
You are the one who started at 0, remember?
— fishfry

I believe the confusion arises from the dual meanings of "start" due to there being two timelines: (1) my timeline as the creator of the story and (2) the timeline of the man running from 0 to 1 within the story. — keystone

The stories are very unhelpful to me. As are timelines.

On my timeline, I start by constructing the entire narrative of him running from 0 to 1. — keystone

What is this '0'? What is this '1'? Define your terms.

The journey is complete from the start. I can make additional cuts to, for example, see him at 0.5. Regardless of what I do, the journey is always complete.
On the running man's timeline, he experiences himself starting at 0, travelling towards 1, and later arriving at 1. — keystone

You are using these funny symbols. I know the usual standard mathematical meaning of those symbols, but you have rejected them in favor of your "top down" idea. So what are these symbols? What if we called them "fish" and "bazooka?" Then nothing at all would be clear, but your logic error would be more obvious

You want to reject standard mathematics but freely use symbols like 0, .5, and 1, without defining them.

Do you see the problem?

I think you're trying to build his journey on his timeline, one point at a time. The runner would indeed believe that limits are required for him to advance to 0.5. I want you to look at it from my timeline (outside of his world), where the journey is already complete. If I want to see where he is at 0.5 I just cut his complete journey in half. Does that clarify things? — keystone

No, since I don't know what 0.5 and "half" mean, in the absence of standard bottom-up math.

Do you see your circularity problem? You want to start by rejecting standard math, but then you won't tell me what these symbols mean in your system.

Unlike supertasks, no magic is required to complete the journey with the top-down view. Assuming you accept the Peano Axioms as a conventional framework, — keystone

Ah. That's quite a lot already, for someone claiming to reject infinite processes and standard bottom-up math.

So you are willing to start with the Peano axioms? Is that your starting place? Then I know what 0 and 1 are, but I'm still not sure about this 0.5 thing.

you're familiar with the concept of succession, which defines progression from 1 to 2 to 3, and so on. This is essentially what I'm applying as well; on the runner's timeline he progresses in succession from 0 to (0,0.5) to 0.5, — keystone

0.5 is not defined by the Peano axioms. What is it?

and so on. Please take note, this particular succession from 0 to 0.5 involves only 2 steps. No limit is required, just as no limits are employed with the Peano Axioms. — keystone

No idea what 0.5 is. But at least after all this you agreed to stipulate the Peano axioms. That's a start. A start from classical, bottom-up math.

I'll save you some trouble and show you how to build out the rational numbers from the Peano axioms. You extend the natural numbers to the integers, then you do a construction called the field of fractions of an integral domain.

I'm not entirely sure if that construction is legit in Peano without the axiom of infinity, but I can live with it.

So after all this, I think you are working in the rational numbers, and 0.5 has its usual meaning. Is that right?

I can live with that. Although the rational numbers are tragically deficient as a continuum. You know that, right? They're full of holes. They're not continuous in the intuitive sense.

'Vastly' is a big word. By quick look-up, the average welder's pay is $22.55/hr, while the average primary school teacher's is $23.44/hr. The teacher starts working life with a $58,000 student loan; the welder gets certification for $475. — Vera Mont

Cherry picking teachers is misleading. A quick Google search on "how much to college graduates earn?" said that they make $50k their first year. "Average college graduate salary" yielded $67,786.

But still, you said you're a communist. Aren't communists supposed to be on the side of the workers? Why should the welder pay the teacher's debts, or anyone else's debts? Why shouldn't everyone pay their own debts? And again, if Congress wants to change that, let them pass a law. The president is not authorized to transfer billions of dollars of student debt to everyone BUT the people who agreed to pay that debt.

You keep saying it's the working class who will be 'burdened' by educating its children, so that they can still work when all the working-class jobs except home renovation and domestic service are automated out of existence. Why do you think poor people's kids shouldn't have a choice of careers? — Vera Mont

Do you still beat your wife?

What kind of question is that? It has nothing to do with anything. You're just changing the subject. And arguing that the working class should assume the debts of the college grads. Some commie you are! You still haven't explained this to me.

I'm the one on the side of the workers. I'm a better commie than you are and I'm not even a commie. I used to be one, then I learned something about the world.

President Biden will announce plans that, if finalized as proposed, would cancel up to $20,000 of the amount a borrower’s balance has grown due to unpaid interest on their loans after entering repayment, regardless of their income. — Vera Mont

The debt is not cancelled. It's transferred to the taxpayers. Can we at least be clear about that?

Low and middle-income borrowers enrolled in the SAVE plan or any other income-driven repayment (IDR) plan would be eligible for the entire amount their balance has grown since entering repayment to be canceled under the Administration’s plans. This group of borrowers includes single borrowers who earn $120,000 or less and married borrowers who earn $240,000 or less. — Vera Mont

Not a nickel of debt is cancelled. It's transferred to the taxpayers. I'll concede that you seem to have paid more attention to the details of the plan than I have.

If Congress wants to pass subsidies for the debt of low-income people, let him do that. But why stop at college debt? Why not transfer everyone's credit card and mortgage debt to the taxpayers as well? After all, isn't home ownership a social good?

As for transferring the tax burden from the elite to the working class - - - ? I guess it depends what newspaper you're reading. — Vera Mont

I read the ones that say Biden is cancelling some student debt. By definition, that excludes non-students, people who didn't go to college and didn't take out student loans. So the non-students pay (via taxes and inflation due to the additional borrowing required to pay off the banks) the legally contracted debt of the students.

What do your newspapers say?

President Biden’s tax cuts cut child poverty in half in 2021 and are saving millions of people an average of about $800 per year in health insurance premiums today. Going forward, in addition to honoring his pledge not to raise taxes on anyone earning less than $400,000 annually, President Biden’s tax plan would cut taxes for middle- and low-income Americans — Vera Mont

Is this a campaign ad?

Another Google quickie revealed that Biden's inflation has cost the average family $8,508 relative to before Biden took office. We could play this game all day. What do Biden's tax cuts have to do with his illegal student loan forgiveness?

You keep defending that one deluded man, and don't care how his co-workers struggle to give their children a chance in a fucked-up capitalist society. — Vera Mont

Sorry, what? What one deluded man am I defending? Whose co-workers? Fuck capitalism, down with the man, eat the rich, up the revolution!! Can you try to focus on the conversation?

I saw a pretty funny sign last night:
"Did anyone think to unplug America and plug it in again?"
The system's been cracking for a long time; all anyone can do, short of smashing it and starting over, is apply patches here and there. — Vera Mont

I'll grant you that Marx's predictions about late-stage capitalism seem to be coming true. We don't actually have much capitalism anymore, we have an oligarchy causing unsustainable inequality leading to a revolution or a cyber totalitarian nightmare. The system's broken. In fact the economy is only being held up by government borrowing and printing at this point. You and I may be in agreement on some things.

Well between the two of you I have no idea what a supertask is anymore.
— fishfry

A supertask is "a countably infinite sequence of operations that occur sequentially within a finite interval of time." — Michael

Well ok, then why don't I complete a supertask when I walk across the room, first going halfway, etc.? Can you distinguish this case from your definition?

What I think about supertasks is:

* Either they are already possible in the sense of Zeno, when I walk across the room; or

* They are physically impossible in currently known physics (because of Planck) but may be possible in future physics, by analogy with previous scientific revolutions; and

* You have not convinced me or even made me understand your reasoning that supertasks are "metaphysically impossible" or that they entail a logical contradiction.

I agree with you. — Ludwig V

I love when people agree with me. It happens so seldom around here :-)

It suits my approach well, in that the existence of the problem is a result of the way it is defined, or not defined. — Ludwig V

I agree with you too!

The walker is on step one, the walker is on step two, etc. So if we define the final state to be that the walker is at the bottom of the stairs, that definition has the virtue of making the walker's sequence continuous.
— fishfry
That's the way ω is defined, isn't it? Although I'm not sure what you mean by "continuous" there.
I still feel uncomfortable, because it does get to the bottom of the stairs by placing a foot on each of the stairs, in sequence. But that's exactly the hypnotism of the way the problem is defined. And if an infinite physical staircase is the scenario, then anything goes.. — Ludwig V

Let me see if I can clarify my point.

In the lamp problem, we have the sequence 0, 1, 0, 1, 0, 1, ... We can "complete" the sequence by defining the state at $\omega$ as 0, or 1, or a plate of spaghetti. In no conceivable completion can the sequence be made continuous, because 0, 1, 0, 1, ... simply does not have a limit.

But in the staircase problem, if 1 is "walker is on the step" and 0 otherwise, then we have the sequence 1, 1, 1, 1, ... which has the limit 1. So 1, the walker is on the step, is the natural state at the end of the sequence.

Does that make sense? The staircase has a natural answer; but the lamp has no natural answer. Any completion whatsoever is as equally bad as any other.

Yes. But I have an obstinate feeling that that fact is a reductio of the process that generated it. So I'm not questioning what you say, but rather what we make of it. — Ludwig V

Right. So why is a lamp circuit that can switch states in arbitrarily small slices of time reasonable, and spaghetti isn't? That's one of the cognitive traps of the lamp problem. IMO the final state is simply not defined by the premises of the problem, AND there is no solution that makes the sequence continuous, therefore spaghetti is as sensible as anything else. And I've convinced myself that this is the solution to the problem.

It may be a bad habit to think of applications of a mathematical process. But that's what's going on with the infinite staircase. So it might be relevant to that.
3 minutes ago — Ludwig V

The staircase is different from the lamp. The walker is on step one, the walker is on step two, etc. So if we define the final state to be that the walker is at the bottom of the stairs, that definition has the virtue of making the walker's sequence continuous. So it's to be preferred over all other possible solutions.

I understand. It's probably best not to comment any further. — Ludwig V

Thanks, I hope that didn't come out too ... however it came out. J6 is a sore point on both sides of the issue. If I said anything at all I'd be inviting discussion so I'll just refrain. Anyway some of the discussion about education policy was outside my area of expertise and interest, so I haven't got much else to say here.

Ok, I think that I finally have learned my lesson now. I will never try to defeat formalism again. Seriously, this was my last attempt. — Tarskian

So glad I could provide some insight. That was an interesting question. Formalism is saved!

I certainly do not believe that mathematics revolves around the correspondence with the physical universe. By "correspondentist", I actually mean: correspondence with a particular designated preexisting abstract Platonic world, such as the natural numbers. — Tarskian

The natural numbers have no physical instantiation as far as we know. Their existence is only abstract, fictional if you will. Or, to a formalist, purely symbolic.

It is beyond question that the axiom of infinity gives us a model of the Peano axioms, but both structures are equally fictional or equally formal.

Mathematical realism is about the independent existence of such Platonic universes. — Tarskian

Ok ... but ZF is a symbolic system. It doesn't talk about things in the real world, only sets, whose existence is entirely formal or fictional or abstract.

If these Platonic universes do not even exist, why try to investigate the correspondence with a particular theory? It only makes sense if they do exist, independent of mathematics or any other theory. — Tarskian

Maybe that's a good question but I'm not sure. ZF exists independently of PA, but both are symbolic axiomatic systems.

Model theory truly believes that the natural numbers exist independently from mathematics or any of its theories. — Tarskian

I'm not qualified to agree or disagree, but it sounds suspect to me. I don't think that the model theorists every say to the set theorists, "I bet you didn't know that sets are as real as cheeseburgers." Nobody ever says that or believes it. The structures studied in model theory generally live in set theory. And there's nothing real about set theory except by virtue of our imagination and symbolic processing.

ps -- Ok I have a sharper response.

The Wiki article on model theory says:

"In mathematical logic, model theory is the study of the relationship between formal theories (a collection of sentences in a formal language expressing statements about a mathematical structure), and their models (those structures in which the statements of the theory hold)"

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

But then when you click on structure, it says"

"In universal algebra and in model theory, a structure consists of a set along with a collection of finitary operations and relations that are defined on it."

https://en.wikipedia.org/wiki/Structure_(mathematical_logic)

So model theory studies the structures that satisfy some axioms; buy the structures themselves are nothing more than formal systems. A set along with a collection of operations and relations.

I think that resolves your concern. One can study a set along with some operations and relations defined on it, without believing such a set is real or has concrete existence or whatever way you are expressing your concern.

In short, the structures can be taken to be every bit as syntactic as the axioms that the structures are models of.

Ok. Perhaps you and Michael could hash this out. He thinks supertasks are metaphysically impossible
— fishfry
Perhaps he does, but he fallaciously keeps submitting cases that need a final step in order to demonstrate the contradiction. I don't. — noAxioms

Well between the two of you I have no idea what a supertask is anymore.

I say they're conditionally physically possible, but the condition is unreasonable. There seems to be a finite number of steps involved for Achilles, and that makes the physical case not a supertask. I cannot prove this. It's an opinion. — noAxioms

I tend to agree with you, that supertasks either (a) may be physically possible via the physics of the future; or (b) are already possible when I go from the living room to the kitchen for a snack, first traversing half the distance, then half of the remaining half, and so forth, and somehow miraculously arriving at my refrigerator. Which keeps things cold in a warm room, in clear violation of the second law of thermodynamics. Truly we live in remarkable times.

Do you have a hard time with 0 being the limit of 1/2, 1/3, 1/4, 1/5, 1/6, ...? It's true that 0 is not a "step", but it's an element of the set {1/2, 1/3, 1/4, 1/5, 1/6, ..., 0}, which is a perfectly valid set.
— Ludwig V
I have no problem with any that.

You can think of 0 as the infinitieth item, but not the infinitieth step.
OK, that's probably a problem. It is treating something that isn't a number as a number. It would suggest a prior element numbered ∞-1. — noAxioms

You believe in limits, you said so. And if you believe even in the very basics of set theory, in the principle that I can always union two sets, then I can adjoin 1 to {1/2, 1/3, 1/4, 1/5, ...} to create the set {1/2, 1/3, 1/4, 1/5, ..., 1}.

It's such a commonplace example, yet you claim to not believe it? Or what is your objection, exactly? It's an infinite sequence. I stuck the number 1 on the end. The entire set is ordered by the usual order on the rational numbers. So why is it troubling you that I called 1 the "infinitieth" member of the ordered set? It's a perfect description of what's going on. And it's a revealing and insightful way to conceptualize the final state of a supertask. Which is why I'm mentioning it so often in this thread.

Even if space is continuous, we can't cut it up or even sensibly talk about it below the Planck length.
But you can traverse the space of that step, even when well below the Planck length. — noAxioms

Only mathematically, In terms of known physics as of this writing, we can not sensibly discuss what might be going on below the Planck length. Maybe space is continuous. Maybe we live in a discrete grid of points -- which would actually resolve Zeno's paradoxes. Maybe something entirely different and not yet imagined is going on. We just don't know.

But you can't say "you can traverse the space of that step, even when well below the Planck length" because there is no evidence, no theory of physics that supports that claim.

In physics, the same way as math, except one isn't required to ponder the physical case since it isn't abstract. One completes the task simply by moving, something an inertial particle can do. The inertial particle is incapable of worrying about the mathematics of the situation. — noAxioms

Well yes, motion is possible. That's one response to Zeno. Not so satisfactory though. Did I complete a supertask when I got up to go to the kitchen for a snack? I have no idea, even though motion through space within an interval of time is an every day occurrence.

The closed unit interval [0,1] has a first point and a last point, has length1, and is made up of 0-length points.
So it does. Zeno's supertask is not a closed interval, but I agree that closed intervals have first and last points. — noAxioms

Ok. I thought you were claiming supertasks had to related to open intervals.

I said that Congress should pass a law funding college costs if that's what they want.
— fishfry
I think you said quite a lot more than that. — Vera Mont

Such as ...?

I'm not aware that the elite had been paying for student loans. Citation? — Vera Mont

Citation? Jeez I don't have to read you the daily newspapers, do I? The college students having their loans "forgiven" aka transferred to the working class that you apparently don't like very much, will out-earn the working class by millions of dollars over their lives.

I am really surprised to see a self-described communist want to burden the working class with the student debt of people who will vastly out-earn them. I wonder if you could address this point.

Did we discuss restructuring taxation at all? I have some views on capital gains, shell corporations, off-shore accounts and price-gauging that wouldn't affect most union members. — Vera Mont

No, I'm trying to keep it simple. Biden's illegal and quite regressive transfer of student debt from students to blue collar workers.

Trashing the welder.
— fishfry
Just that one. He probably beats his wife and votes for T***p, too. — Vera Mont

With commies like you trying to saddle him with billions in debt, it's no surprise. You just explained Trump's popularity. The left's abandonment of the working class has a lot to do with it.

I don't think you've done anything at all. — Vera Mont

LOL. Ok I guess we're done. Nice chatting with you. Hope you'll give some private thought to why you are defending the transfer of debt to the working class, whom the communists are supposed to have an affinity for. But that was 50 years ago, wasn't it. Now the left loves the deep state, loves the intel agencies, loves the wars, and hates the working class.

That's why the welder loves Trump. Because the Democratic party and apparently even the communists stopped caring long ago.

I'm sure there are other ways to define the ordering of rational numbers, that's just my favorite. — keystone

So you believe in the rational numbers? But then the reals are easily constructed from the rationals as Dedekind cuts or equivalence classes of Cauchy sequences. If you believe in the rationals you have to believe in the reals.

I thought I twice answered your question. Let me try again. What you don't seem to appreciate is that with the top-down view we begin with the journey already complete so halving the journey is no problem. If we already got to 1, then getting to 0.5 is no problem. You can't seem to get your mind out of the bottom-up view where we construct the journey from points, which indeed requires limits. — keystone

You are the one who started at 0, then got to (0, .5), and then magically completed a limiting process to get to .5. I ask again, how is that accomplished?

You are the one who started at 0, remember?

I don't believe I've said anything to lead you to believe I'm against education.
— fishfry
Only for people who can't afford it. — Vera Mont

But I did not say that. I said that Congress should pass a law funding college costs if that's what they want. Biden's action is illegal. And as a self-described communist, I'm surprised to see you cheering on the transfer of billions of dollars in debt from the elite to the working class. You sure you're a commie? Or are all the commies elitists these days? That's what it seems like.

You said the welders militarized the police.
— fishfry
No i didn't. I said
That welder who'd rather see his taxes go toward militarizing the police is doing his family no favours. — Vera Mont

"There you go again," as Reagan once said to Jimmy Carter. Trashing the welder. You don't think much of the working class? You sure you're a commie? I mean you say you are, but your words say otherwise.

Don't tell me there isn't one single yahoo in the welder's union who wouldn't rather beef up the police than give some pansy a degree in social work. There is. And he's an idiot. — Vera Mont

Yahoos. So either you're a commie with disdain for the working class, or else communism is now a faddish pastime of the elite. Which is exactly what it is these days, at least in the US.

No, I'm anti representing all working class people as thinking like you. — Vera Mont

I couldn't actually parse that except that I must have done something bad.

I take it you're not a fan of analogies. — keystone

I like analogies fine. I don't understand any of yours. I thought we were making progress on at least having the same conversation when we were traversing the unit interval. Instead of engaging you're changing the subject.

0 and 0.5 have distinct positions on the Stern-Brocot tree. — keystone

You're taking that as fundamental?

I like football but these picture posts aren't doing much for me. We were at least having the same conversation about getting from 0 to 1 on the real line. Then you said you don't believe in the real numbers, and then you declined to respond when I asked you twice how you get from (0, .5) to .5 without invoking a limiting process. And now you're changing the subject.

Model theory makes anti-realist views unsustainable. — Tarskian

I don't see how that is. Take as an example the Peano axioms for the natural numbers. Do we have a model of them? Yes, namely $\mathbb N$, the smallest inductive set guaranteed by the axiom of infinity in ZF set theory.

But the latter is just as fictional as the former, is it not? There's no empirical proof of the existence of infinite sets. They're a mathematical abstraction.

It seems to me that models are often purely abstract mathematical entities. One can take a purely formalist view of ZF for example. There are no sets in the real world in the sense of set theory. Show me the set containing the empty set and the set containing the empty set, which is better known by its more familiar name, 2.

Sure. But a society needs a variety of skills. And it needs to recognize the need for education, and the need for recognition of talent, in whatever class, whether they can play basketball or not, whether they can afford a huge debt-load or not. — Vera Mont

I don't believe I've said anything to lead you to believe I'm against education. I'm against Biden's illegal, election year transfer of lawful debts from the people who signed for them. to the taxpayers. Have I said anything more than that?

You're lecturing a communist about the working-class and elitism? — Vera Mont

You prefer Stalin? Or Mao? You endorse mass murder? Some of what you said sounded a bit elitist.

On many of the wrong things, because they're bound by old obligations, treaties, contracts, attitudes and fears. Investing in youth is one of the right things it should be spending on. — Vera Mont

If the US government wants to "invest in youth," as you call it, why doesn't Congress pass a law forgiving all student debt? They didn't do that. Biden has no actual authority to do it. The Supreme court has already ruled on his previous debt bailout.

I am not arguing about investment in education. I'm arguing against Biden's election year bailout of college students at the expense of the working class, whose interests you should in theory be defending.

If you are a communist, you should be agreeing with me about this! Right? Workers of the world unite, take on the debt burden of the people who will out-earn you by millions over their lives. That the story you're going with?

Don't they always? Then, for about 20 years, the ultra-rich keep their greed in check and their profile low. Then they start buying up politicians and smaller businesses and countries again. — Vera Mont

ok

It's not the welders who have militarized the police.
— fishfry
Of course it isn't. But that's where their taxes go anyway, because the people who have lots of property want it protected at public expense. — Vera Mont

You said the welders militarized the police. I pointed out that's not true.

You're quite anti-worker for a communist. Am I out of date on my understanding of communist ideology? Are today's communists all for the bourgoisie?

I'm just getting around to responding to a few of your earlier comments:

As it happens I hate that stupid movie. It's a kung-fu flick with silly pretensions to pseudo-intellectuality. Also someone did the calculation and it turns out that humans make lousy batteries. Very inefficient. Where is the line between your indulging yourself, and your trying to communicate a clear idea to me?
— fishfry
Wow, it's one of my favourite films. To each their own, I suppose. It seems we view things quite differently in several respects. That's exactly why I find this conversation so valuable. — keystone

Even taken at face value, I fail to understand how posting stills from the movie relates to anything we're discussing. And like I said, humans make lousy batteries. So the premise of the film is wrong. I agree with it as a metaphor for media and government treating us all as tax cattle.

Like a triangular section of the plane? Why?
— fishfry
JGill noted that using x and y for my upper/lower bounds was confusing. I think that's why you were confused with my earlier post. Hopefully using a and b is less misleading. — keystone

a and b is less confusing than x and y? I better go back and re-read the thread.

I'm very glad I can help. What is the digital rain? Do you remember the Church of the Cathode Ray from the movie Videodrome?
— fishfry
I was suggesting that our discussion around topological metric spaces has warmed me up to the idea of sets being fundamental. I now believe that, if there is merit to a top-down view of mathematics, that is will be described using sets. I certainly didn't hold that view at the beginning of our conversation. I didn't watch Videodrome, it was a little before my time. — keystone

Great flick though the plot gets a little muddle in the second half. Classic Cronenberg.

As a number, pi, is inseparably tied to actual infinity, — keystone

Pi is a computable real number and only encodes a finite amount of information.

Please stop calling it a topological metric space just as I don't call my cat my cat mammal.
— fishfry
What should I call it? — keystone

A metric space. A metric space is already a topological space, as a cat is already a mammal.

But why denote the point 0 as [0,0]? Isn't that obfuscatory and confusing?
— fishfry
I did that to facilitate the straightforward definition of the metric. If you permit me to work within a metric space without necessitating an explicit definition of the metric, then I will designate it as point 0. — keystone

Calling 0 by the name 0 would be far less confusing. Who was it that said that when discussing transcendental matters, be transcendentally clear. Looked it up, Descartes. Smart guy. You have the burden of being as clear as you can possibly be.

If you mean what's mathematically called a path, I'm fine with that.
— fishfry
I'm talking about a top-down analogue to (bottom-up) paths. By this I mean that (bottom-up) paths are defined using points (real numbers) whereas I'm defining the (top-down) 'path' using continua. I would like to use the term 'path' if you permit me to use it without implying the existence of R. — keystone

If you deny the real numbers then I have no idea what 0 and .5 are, since they are real numbers. What do those symbols mean?

How do you get from (0,0.5) to [0.5,0.5]? …Mathematically, you take a limit.
— fishfry
Let me describe both the bottom-up view and the top-down view.

Bottom-up view
The journey from point 0 to point 0.5 can be constructed as follows:
Step 1: 1/4 = 0.25
Step 2: 1/4 + 1/8 = 0.375
Step 3: 1/4 + 1/8 + 1/16 = 0.4375
Step 4: 1/4 + 1/8 + 1/16 + 1/32 = 0.4688
Step 5: 1/4 + 1/8 + 1/16 + 1/32 + 1/64 = 0.4844
Step 6: 1/4 + 1/8 + 1/16 + 1/32 + 1/64 + 1/128 = 0.4922
…

Along this journey there is no finite step where we arrive at precisely 0.5. This approach requires something like a 'step omega' and to get to 0.5 requires a limit to 'jump' the gap.

Top-down view
We begin with the completed journey from point 0 to point 0.5. Some versions of how the journey can be decomposed are as follows:
Decomposition version 1: 1/2 = 0.5
Decomposition version 2: 1/4 + 1/4 = 0.5
Decomposition version 3: 1/4 + 1/8 + 1/8 = 0.5
Decomposition version 4: 1/4 + 1/8 + 1/16 + 1/16 = 0.5
Decomposition version 5: 1/4 + 1/8 + 1/16 + 1/32 + 1/32 = 0.5
Decomposition version 6: 1/4 + 1/8 + 1/16 + 1/32 + 1/64 + 1/64 = 0.5
…

The various versions correspond to how we might chose to make cuts.
For example, the journey in decomposition version 2 is [0,0] U (0,1/4) U [1/4,1/4] U (1/4,1/2) U [1/2,1/2].
Regardless of how many cuts we make (i.e. regardless of what version we're looking at), the journey is always complete. No limits are required. Limits are only required to make the top-down view equivalent to the bottom-up view (i.e. decomposition version omega = step omega).

The confusion seems to stem from you viewing the interval (0, 0.5) as an infinite collection of points (naturally, since that is a bottom-up perspective of an interval). However, from a top-down perspective, the interval (0, 0.5) represents a single object - a continuum (perhaps I should return to calling it a k-interval to avoid confusion). While this continuum indeed has the potential to be subdivided infinitely (much like an object can potentially have holes), until actual cuts are made, we cannot assert the existence of actually infinite discrete points.

Going back to the set {0 , (0,0.5) , 0.5 , (0.5,1) , 1} , all that exists are 3 points and 2 continua and for a continuous journey we advance through them in this order proceeding from one step to another without taking limits:

Step 1: Start at point 0.
Step 2: Travel the continuum (0,0.5)
Step 3: Arrive at point 0.5.
Step 4: Travel the continuum (0.5,1)
Step 5: Arrive at point 1. — keystone

Okay, I've thought about this further and I think you're right! Do the following 5 intervals make more sense? None of them are empty anymore. For points, let's use closed intervals.

Interval 1: [0,0]
Interval 2: (0,0.5)
Interval 3: [0.5,0.5]
Interval 4: (0.5,1)
Interval 5: [1,1] — keystone

I apologize. All this makes my eyes glaze. It makes no sense to me.

And since you denied believing in the real numbers, I don't know what those symbols mean. Perhaps you can start there.

But look. I asked you this last time.

"Step 2: Travel the continuum (0,0.5)
Step 3: Arrive at point 0.5."

How do you get to .5 fom (0, .5)? Don't you have to take a limit? This is an important question. You seem to be implicitly willing to take limits, while denying the real numbers. I see that as a problem.

The sequence itself has no last item. But the "augmented sequence," if you call it that, does. We can simply stick the limit at the end.
— fishfry
I think that's all right. When I walk a mile, I start a potentially infinite series of paces. When I have done (approximately) 1,760 of them, I stop. The fact that the 1,760th of them is the last one is, from the point of the view of the sequence, arbitrary, not included in the sequence . The sequence itself could continue, but doesn't. — Ludwig V

I was making my point about mathematical convergent sequences. Don't know whether it strictly applies to walking.

I suspect a nation of welders would starve to death pretty fast. — Vera Mont

A nation of farmers would live a lot longer than a nation of comparative literature majors, I'm sure you agree. The trades are "real work." Tradesmen built the college buildings, they operate the plumbing and the electricity and haul the trash. Without them, the lotus eaters would not be able to function at all. That's the parable of the Eloi and the Morlocks. The Eloi forgot how to make or build or create or maintain. All that's done for them by the dirty, underground, loathsome Morlocks. A metaphor for the attitude of the elite towards the working class today. And that's why it's good to recall that the Morlocks eat the Eloi from time to time.

And a debt-driven society will inevitably collapse under the burden. — Vera Mont

The US is $35T in debt and still spending like a drunken sailor. Though as Ronald Reagan quipped, at least the drunken sailor is spending his own money.

Student loans - agreed to by unemployable youth who hope for a future, come at 5-15% interest. They won't earn enough to live on, let alone pay off $50, -100, 000 for years after they graduate, so the interest just keeps on accumulating. — Vera Mont

There's a reason the cost of school keeps going up: government guaranteed student loans. Under that system the schools have no incentive to keep costs down. The banks are willing to lend to people who will never pay them back, since the taxpayers backstop the loans. The result is massive inflation in college costs, far outpacing the inflation rate of other goods and services. And now the students don't have to pay the loans back at all, and the $35T-indebted taxpayers get loaded up with still more debt.

When this whole thing crashes everyone's going to go, "Oh how did we let it get this bad?" But till that day ... party on. Stick the grandkids with the bill. It's really a depraved thing, what the US is doing with its own economy, running up a tab it can never pay.

So the most ambitious and clever of them will vie for the lucrative corporate and money-shuffling jobs that do nothing for the population - because they can't afford to work in low-paid public service or helping professions. — Vera Mont

Yes, and this is my point. Government-backed student loans that then don't even need to be paid are a moral hazard and a great distortion of the labor market. Better to give each high school graduate $100k and let them spend it in college or trade school or party it all away. It would be a better system than what we have now.

That welder who'd rather see his taxes go toward militarizing the police is doing his family no favours. — Vera Mont

It's not the welders who have militarized the police. That trend started when the big-name Democrats like Pelosi, Hillary Clinton, Schumer, and Biden (as Senator) signed on to Bush's illegal wars. When the wars wound down, all that military hardware was given to local police departments. We then saw the MRAPs and other military hardware on the streets of the US during Occupy and later at Ferguson. It wasn't the welders. They don't set government policy.

I said no such thing!! If you like, you can think of the limit as being the ∞-th item.
— fishfry

There is an ∞-th item, namely the limit of the sequence.

The sequence itself has no last item. But the "augmented sequence," if you call it that, does. We can simply stick the limit at the end.
— fishfry

then 1 may be sensibly taken as the ∞-th item, or as I've been calling it, the item at ω
— fishfry

Then you say.

If it is indeed accomplishing an infinite amount of steps, is there not a step where the sequence gives us 1?
— Lionino

No.
— fishfry

Is there not a contrast between these two sets of statements? — Lionino

No. Consider the sequence 1, 1/2, 1/3, 1/4, 1/5, 1/6, ...

It has the limit 0.

We may form the ordered set {1, 1/2, 1/3, 1/4, 1/5, 1/6, ..., 0}. It's a perfectly sensible set.

In this context 0 is the largest element in the set. It's the final "item" if you like. But 0 is not any step in the sequence 1, 1/2, 1/3, 1/4, 1/5, 1/6, ..., and that sequence has no last step.

Is my use of the words step and item more clear?

We are applying mathematics not just to this physical world but to any possible world where the physics could be different, and for that we discuss what the mathematics means in the world — as it is necessary that 1+1=2 so that everytime you take one of something and one again you end up with two. — Lionino

Ok. But one has to be careful of applying math to the world, this one or any other. Physicists typically model time as a real number, but there's no evidence that time is a continuum as the real numbers are. So math gets applied to physics heuristically or pragmatically, and not metaphysically. We model time using the real numbers because it's handy and gets us results, not because we actually believe time is like the real numbers.

I don't see how you could count all the natural numbers by saying them out loud or writing them down. Is this under dispute?
— fishfry
No. Nobody seem to have suggested that was possible. It simply isn't a supertask. — noAxioms

Ok. @Michael has been using that as an example of a supertask so I can't say. I haven't studied them much.

Do you mean the fact that I can walk a city block in finite time even though I had to pass through 1/2, 3/4, etc? I agree with you, that's a mystery to me.
Yes, I mean that, and it's not a mystery to me. If spacetime is continuous, then it's an example of a physical supertask and there's no contradiction in it. — noAxioms

Ok. Perhaps you and @Michael could hash this out. He thinks supertasks are metaphysically impossible, and you think they're everyday occurrences. I'm agnostic on the matter except to say that I don't think they're metaphysically impossible, whether they're physically possible or not.

No, the lamp changes things. It introduces a contradiction by attempting to measure a nonexistent thing. That in itself is fine, but the output of a non-measurement is undefined. — noAxioms

The state of the lamp is defined at each of the times 1/2, 3/4, 7/8, ... but it's not defined at 1.

Like any other function defined at some elements of a set but not others, I am free to define it any way I like.

I looked up [Bernadete's Paradox of the God], didn't seem to find a definitive version.
Nicely stated by Michael in reply 30, top post of page 2 if you get 30 per page like I do. — noAxioms

Thanks I'll check it out.

Ah the ping pong balls. Don't know. I seem to remember it makes a difference as to whether they're numbered or not.
It's important to the demonstration of the jar being empty, so yes, it makes a difference. — noAxioms

Something went wrong with the quoting when I quoted your post. Anyway ... yes the ping pong balls. I have no opinion about that one.

The outcome seems undefined if they're not numbered since no bijection can be assigned, They don't have to have a number written on them, they just need to be idenfifed, perhaps by placing them in order in the jar, which is a 1-ball wide linear pipe where you remove them from the bottom. — noAxioms

ok

It nicely illustrates that ∞*9 is not larger than ∞, and so there's no reason to suggest that the jar shouldn't be empty after the completion of the supertask. Again, it seems that any argument against this relies on a fallacious assumption of a last step that sooo many people are making in this topic. — noAxioms

I'll agree that the subject of omega sequence paradoxes is full of fallacious assumptions and confused thinking.

So I believe I've been trying to get across the opposite of what you thought I said. There is an ∞-th item, namely the limit of the sequence.
— fishfry
That can't be a step, since every step in a supertask is followed by more steps, and that one isn't. I have a hard time with this ∞-th step. — noAxioms

I say "item" and you change the word to "step," changing my meaning. I agree, it's not a step in a sequence. It's an item in a set.

Do you have a hard time with 0 being the limit of 1/2, 1/3, 1/4, 1/5, 1/6, ...? It's true that 0 is not a "step", but it's an element of the set {1/2, 1/3, 1/4, 1/5, 1/6, ..., 0}, which is a perfectly valid set. You can think of 0 as the infinitieth item in an ordered set, but not the infinitieth step of a sequence.

The cutting up of the path into infinite steps was already a mathematical exercise. The fact that the physical space can be thus meaningfully cut up is true if the space is continuous. That latter one is the only barrier, since it probably isn't meaningfully, despite all our naïve observations about the nice neat grid of the chessboard. — noAxioms

Even if space is continuous, we can't cut it up or even sensibly talk about it below the Planck length. With our present understanding of the limitations of physics, the question of the ultimate nature of space is metaphysics and not physics.

In math? Via the standard limiting process. In physics? I don't know,
— fishfry
In physics, the same way as math, except one isn't required to ponder the physical case since it isn't abstract. One completes the task simply by moving, something an inertial particle can do. The inertial particle is incapable of worrying about the mathematics of the situation. — noAxioms

Yes ok ... math and physics are human inventions that bear some mysterious relation to reality. I agree with that, if that's what you meant.

How do dimensionless points form lines and planes and solids?
— fishfry
Mathematics: by not having a last one (or adjacent ones even). — noAxioms

Not sure what you mean. The closed unit interval [0,1] has a first point and a last point, has length1, and is made up of 0-length points.

I find this very confusing. I take your point about abstraction. But I find that abstraction can create confusion, because it persuades us to focus on similarities and neglect differences. My reaction here is to pay attention to the difference between these kinds of infinite series. It's not meant to contradict the abstraction. — Ludwig V

The two sets in question have the same order type, denoted $\omega + 1$. That's mathematically true, and it's all that's relevant to these two examples. I'm not sure what's gained by focussing on the differences. $\omega$ is the limit of 1, 2, 3, ... in exactly the same sense that 1 is the limit of 1/2,, 3/4, 7/8, ..., under the more general topological definition of a limit needed to defined limits among the ordinal numbers.

It's a thought experiment. There are no infinite staircases.
— fishfry
Exactly. So it isn't about physics. But it isn't about mathematics either. So it seems to me an exercise in imagination, and that provides a magic wand. — Ludwig V

Yes ok, so the coach can turn into a pumpkin and the lamp can turn into a plate of spaghetti. Are you agreeing with me on that point?

Aha. You'd have to ask those who care so much. I think they only show that underspecified problems can have arbitrary answers. But others see deeper meanings.
— fishfry
Deep? or Deepity? (RIP Dennett) — Ludwig V

RIP.

Yes. Euclid (or Euclidean geometry at least) starts from a foundation - axioms and definitions. But they are an extension of our common sense processes of measuring things. (You can understand more accurate and less accurate measurements.) Extend this without limit - Hey Presto! dimensionless points! That is, to understand what a point is, you have to start from lines and planes and solids and our practice of measuring them and establishing locations. I find that quite satisfying. Start with the practical world, generate a mathematics, take it back to the practical world. (Yes, I do think that actual practice in the real world is more fundamental than logic.)
Once you define geometrical points in that context, there is no difficulty about passing or crossing an infinite number of points. (But the converging series does not consist of points, but of lengths, which are components.) — Ludwig V

Ok

So there's something interesting going on.
— fishfry
My supervisor used to say that when he got really excited, which was not often. — Ludwig V

[/quote]

yes

when that breaks down, the system is in serious danger. (Trump!) — Ludwig V

Actually it's Biden who engaged in election denialism this week when he gave the Presidential medal of freedom to Al Gore, in part for not disputing the 2000 election that he allegedly won. In truth, Gore did lose that election, and he also did dispute the hell out of it, all the way up to the Supreme court.

I'll bow out of this thread now. I engage in partisan politics on this forum on a very limited basis these days. I prefer not to take this bait any more than necessary. Appreciate the insightful chat about education policy.

I'm defining the journey from 0 to 1 using the following 5 intervals:
Interval 1: [0,0]
Interval 2: (0,0.5)
Interval 3: [0.5,0.5]
Interval 4: (0.5,1)
Interval 5: [1,1]

1,3, and 5 correspond to points.
2 and 4 correspond to continua. — keystone

Ok, that at least makes sense. But why denote the point 0 as [0,0]? Isn't that obfuscatory and confusing?

To me, it's obvious that the union of the above 5 intervals completely describes the journey from 0 to 1. Do you agree? — keystone

The union covers the closed unit interval. I don't know what a journey is. If you mean what's mathematically called a path, I'm fine with that. If you mean that you can get from 0 to 1 by first going from 0 to 0, then from 0 to not quite .5, then jumping to .5, then jumping from just above .5 to just before 1 ...

How does all this jumping take place? To get from (0, .5) to 1 involves taking a limit. How do you do that in your journey-mobile?

I'm using intervals to describe all 5 parts of the journey because I want to use intervals in my topological metric space. Let me go ahead and do this... — keystone

Please stop calling it a topological metric space just as I don't call my cat my cat mammal.

Set M is has following ordered pairs (not intervals) as elements:
Ordered pair 1: (0,0)
Ordered pair 2: (0,0.5)
Ordered pair 3: (0.5,0.5)
Ordered pair 4: (0.5,1)
Ordered pair 5: (1,1) — keystone

How do you accomplish those limit jumps? As a set-theoretic union they're fine, but as a "journey" you have a problem. How do you get from an open interval to its limit?

[EDITED FOLLOWING ORDERED PAIRS VARIABLE LETTERS FROM (X,Y) TO (A,B) ACCORDING TO JGILL'S LATER FEEDBACK]

As I mentioned before, the metric between ordered pairs (a1,b1) and (a2,b2) is defined as follows:
d((a1,b1),(a2,b2)) = | (a1+b1)/2 - (a2+b2)/2 |

This metric essentially measures the distance between the midpoints of two intervals. Hopefully this clarifies why I chose to represent points as intervals. — keystone

I'll address the real numbers once we've clarified the topics above. It's not feasible for me to provide a satisfactory response if we're not in agreement on these preliminary matters. — keystone

Just wondering about those jumps to the limit.

Okay, I've thought about this further and I think you're right! Do the following 5 intervals make more sense? None of them are empty anymore. For points, let's use closed intervals.

Interval 1: [0,0]
Interval 2: (0,0.5)
Interval 3: [0.5,0.5]
Interval 4: (0.5,1)
Interval 5: [1,1] — keystone

How do you get from (0,0.5) to [0.5,0.5]?

Do you understand that any point in [0.5,0.5] is a nonzero distance from .5? How do you jump that gap?

Mathematically, you take a limit.

But how does your "journey" take a limit?

Churchill does quote it, but doesn't take credit for it. Also, there are various forms of it. See Quote Investigator. — Ludwig V

I believe Churchill also said that the greatest argument against democracy is a five minute conversation with the average voter. :-)

I thought I had already said that I don't have a problem with that. When I said that politics is a messy business, best not really be conducted in public, I was also accepting that it was a bribe to voters. All democracies do that - it's an inevitable outcome of the system. Non-democratic governments do it as well. Politicians have to keep their supporters sweet. I'm not even saying it is right, or all right, just that it always happens. — Ludwig V

Biden's bailout is particularly egregious, being flagrantly illegal in the first place.

Each country has its own system. In the UK, the "trades" like welding and pipefitting, do get government support - and this is a "right-wing" government. See Skills for careers. Some people regard this as a blatant subsidy for employers, who should be paying. But there are complications.
Higher-level professions depend on degree-level courses, and these get student loans. But these are repaid on a sliding scale, dependent on you income. (Effectively, it's an additional income tax). The Government assumes that 35% to 40% of the total will never be repaid. There's your forgiveness, but sanctioned by Parliament.
Re-training is more of a problem. — Ludwig V

That's very interesting! I wouldn't mind government subsidies if they're fairly distributed among the college and non-college individuals.

Is there a name for the logical fallacy that "P is repugnant, therefore not-P."
— fishfry

Willard von Orman Quine :razz: — Lionino

I'm afraid I don't get the joke.