Now it is also true that 4/0=∞ and 9.7181=∞. And with a little more leg work I shall demonstrate that all numbers are actually equal to each other. Multiplicity is mere illusion, a result of the Fall and Adam's sin. — Count Timothy von Icarus

I look forward to mankind's return to the Garden of Eden.

Thank you very much for those.

If I've understood, your argument shows what the sum of the infinite series is.

Right?

Oh I see what happened. Ludwig brought up the old .999... = 1 chestnut in the staircase thread, and it apparently got moved over here to revivify this four year old thread. — fishfry

It probably saves time and energy. Actually, you mentioned it and I got curious. I'm afraid I innocently asked a question and set off a land-mine.

What number can possibly get between ALL the terms of that sequence, and the number 1? — fishfry

Well, if I've understood how this works, there is a number that gets between each element of the sequence - the next element in the sequence - and is there is no last element of the sequence. So there is no answer to your question.

However, it is also true that 1 is the sum of the infinite series 0.999... - and therefore the limit.
But an infinite series never reaches its limit. To put it another way, "=" in this context (an infinite series) does not mean what it usually means.

Ah, the so-called non-existing dog is the one who doesn’t anything at all. I get it now. But I assumed every dog ate at least a bit. — javi2541997

Is there a non-existing dog? If there is, it doesn't exist. If there isn't, it doesn't exist.

Yes, this is how I see the tricky game. If I'm not mistaken, the dog who eats less than the preceding dog would be represented by 0.00000000…, and so on. However, this dog does exist. It consumes something, even when it is infimum. — javi2541997

Exactly.

Sorry, I was foolish in trying to follow usual norms when infinity is involved. :sweat: — javi2541997

I've been bitten by that infinity more times than I can count. All common sense has to go out the window. It is possible to get used to it.

a=0.999...,
10a=9.999...,
10a-a=9,
9a=9,
a=1 therefore 0.999...=1 — Lionino

I do appreciate your help.

But this doesn't seem to work with other similar sequences, such as 0.333... or 0.444... or 0.1212....
What have I got wrong?

By accepting transcendental dogs and their transcendental food, I argue that you have already accepted (perhaps unintentionally) the existence of Zeno's least eating dog. — ssu

The transcendental food was a joke, playing on the absurdity of transcendental dogs. I must be more careful about jokes.

What is the criterion for Zeno's least eating dog?
Is there an infinite number of dogs?
What is the difference between transcendental dogs and ordinary dogs?

The one at the top (the dog who eats the most) and the one at the bottom (the dog who eats the least). — javi2541997

That is only possible if there is a finite number of dogs.
There cannot be a dog that eats the most - there's bound to be another one that eats more. Similarly for the dog that eats the least. Infinity doesn't follow the normal rules.

Honestly, I think those two are always ‘there’ but it is a mistake to try to identify them with numbers. — javi2541997

Well, strictly speaking they are identified by the amount of food they eat, which determines their position in the line.
The numbers identify their position in the line.
So, since they are identical in every way, apart from the amount of food they eat, there is no other way to identify them.
It is easy to think that they must exist, but if the line is infinite, any specified dog has another dog after it.

As I stated to Ludwig V, just having finite, but transcendental numbers like π or e that aren't Constructible numbers already gives the problem of Zeno's dogs, even if we would dismiss the two Zeno's dogs mentioned. — ssu

I don't know the math well enough to be sure, but I think it is possible to place numbers like π or sqrt2 in order among the natural numbers. So every dog will have a different place in the order, depending on how much they eat. So dogs numbered π etc. will be like every other dog in having a number assigned according to how much they eat. Each dog will be different from every other dog and each dog will be the same as every other dog. It depends how you look at it.

I though it might help to quote the rules again:-

1. The dogs are totally similar in every way except that every dog eats a different quantity of food. All the dogs eat the same food, which is divisible and there is enough of it for every dog. — ssu

Once these were put into the line, then came the dogs which ate quantities between these dogs. — ssu

As all dogs do eat something, we have a problem with the non-existent dog that doesn't eat anything, — ssu

I don't recall mentioning any non-existent dogs, nor any that don't eat anything.

Well, a dog eating ⅚ of Plato's dog's food amount isn't either a natural number, so would you deny it to be a dog?
I would not deny it to be a dog and I would be happy to assign a natural number to it depending on where it comes in the ordering.
— ssu

So, let’s say, there is a dog who eats 15 pieces of meat, and there is another dog who eats only 0.0001 pieces of that meat. — javi2541997

I didn't realize, though I should have done, that you are placing the dogs in a single continuous order. But you have defined two infinite sequences, with a common origin. So the start of your Grand Order is not defined, any more than the finish. Your ordering means you have to start from a dog that you cannot identify.

And what about transcendental dogs? They are finite, but the dog that eats π amount compared to Plato's dog? — ssu

You didn't mention them. In any case, they would naturally eat transcendental food - not being able to digest natural food. As for the dog that eats π amount of food, it will have its place in the order, so there's no problem.

Then again, one could reject that the equation for the sum applies. The equation of the infinite sum relies on the notion of limit, and it is the notion of limit that is at play on the 0.999... debate. — Lionino

I can see that point. I didn't look at the issue in the light of infinite series or take on board that it was a question of the sum of an infinite series. I apologize for the distraction.

There is
a=0.999...,
10a=9.999...,
10a-a=9,
9a=9,
a=1 therefore 0.999...=1 — Lionino

That's very neat.

Everybody agrees that mathematics applies to the physical world, but nominalists will broadly say that 2+2=4 is not about the world, so it is not true of it. — Lionino

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.

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. — fishfry

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.

1 is a perfectly sensible probability. — 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.

The argument shows that the premises entail a contradiction, so at least one of the premises must be rejected. — TonesInDeepFreeze

Which one do you think should be rejected?

3×13=1 and 3×0.333...=0.999... — Michael

That makes 0.999999..... = 1 just an illusion created by the notation you have decided to use. It is not a proof. In my opinion. You might have a different idea of what a proof is.

On the other hand, it does show that looking at a problem another way might show that the problem is an illusion. But that would be philosophy.

I'll refer you to this: — TonesInDeepFreeze

I'm deeply flattered. But that is far too much for me to grasp in less than a month or two.

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. — 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.

The argument shows that the premises entail a contradiction, so at least one of the premises must be rejected. — 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.

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.* — 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.

This issue was actually resolved a long time ago by Aristotle, — Metaphysician Undercover

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.

The unintelligibility is due to a thing's matter or potential. — Metaphysician Undercover

I think it would be more accurate to say "The apparent unintelligibility is due to a thing's matter or potential."

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. — 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.

As what may or may not be, "potential" violates the law of excluded middle. — 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.

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

Who says anything about probability when merely mentioning that .9... = 1. — 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.

we prove that .9... = 1. — TonesInDeepFreeze

That's interesting. Can you refer me to a source?

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. — 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.

And actual infinity is the completed infinity. — ssu

Forgive my stupidity, but I don't understand what a completed infinity is.

Notice in the story Athena, the goddess of wisdom, might very well know the answer as she did use the two philosophers for amusement for the other gods. — ssu

Well, it's your story. You are the only person who can provide an answer.

Even in the story Zeno is well aware of this. — ssu

But back to the story: Then doesn't that ω in the story relate to distinct dog? You even referred yourself of ω being a number. — ssu

A transfinite number isn't a natural number, so it doesn't get attached to (aligned with) a dog. Nor could it be.

First of all, notice that ω here refers to the largest Ordinal number. — ssu

I was careful to notice that - and. at least by implication, the cardinal numbers.

you put all the dogs that food amount is exactly divisible by dog 1's food (let's call them natural dogs) in a line from smaller to bigger — ssu

That will take you, and even the gods, an infinite time. But I guess Plato, Zeno and certainly the gods, have that amount of time available, and are bored.

start counting the dog line from their places on the line, from the first, second, third, fourth... and then get to infinity in the form of ω. — ssu

You can start, but you can't finish in less than infinite time. And even Plato, Zeno and the gods will be bored by the time they get to the end of a second infinite count.

Well, you already referred to completed infinity or actual infinity with the example of ω as that is Cantorian set theory. — ssu

If you choose to call ω completed or actual, that's your choice. I can't work out what you mean. I don't know enough to comment on Cantorian set theory.

Oh I see. I prefer shorter posts so I don't get lost in the quoting! — fishfry

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.

With supertasks? I don't think so. — fishfry

I didn't mean to imply that they were living together. That would be .... interestingly mnd-boggling.

See any .999... = 1 debate. — 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.

Peano arithmetic is potential and the axiom of infinity gives you a completed infinity. — fishfry

Since my earlier comment on this,

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. — 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.

The real numbers are the completion of all the sequences of rationals. That's how we conceptualize the reals. — 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.

Math is outside of time. It doesn't describe or talk about time, though it can be used by physicists to model time. — 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.

Right. Trains are physical objects. Numbers in a sequence are mathematical abstractions. They don't live in the physical world. — fishfry

Yes. But there are complications. How does math apply to the physical world?

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. — 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?

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. — fishfry

As I said before there are a number of ways to describe this. They're all a bit weird.

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. — 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.

We can think of this as a FUNCTION that inputs a natural number 1, 2, 3, ... and outputs 1/(2 to the power of n). — 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.

rC3: The lamp is neither Off nor On at 12:00. Contradicts rP1. — 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.

if mathematics is true of the physical world too or rather only applies to it — Lionino

I don't get the difference. If mathematics applies to the physical world, surely it is true of it?

Euclidean geometry applies to a car going from the theater to the restaurant (the surface of the city is flat), non-Euclidean to an airplane going around the Earth (spherical geometry) or things interacting in space-time (hyperbolic geometry). — Lionino

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.

But already Zeno identified two dogs that eat differently than their dogs. — L'éléphant

That's no great trick. Every dog eats differently than all the other dogs.

But the fact remains that there is the dog the eats the most and the dog that eats the least. — L'éléphant

There's an ambiguity in the ordinary use of these superlatives which means they cannot be meaningfully applied in the context of a infinite sequence.

I assume that we can take Plato's dog as dog 0, and allocate the natural numbers with the dogs that eat more than Plato's dog. (And similarly with the dogs that eat less than Plato's dog. Yes?

The largest natural number is the number that is larger than all the other natural numbers and has no natural number that is larger than it. But every natural number has a natural number larger than it. So there is no largest natural number. That follows from the definition of infinity.

It looks as if you are not aware of how the mathematics works in this context. Forgive me if I'm wrong.

There is a number that is larger than every natural number.
That number is ω, which is the lowest ordinal transfinite number, which is defined as the limit of the sequence of the natural numbers.

See Wikipedia - Transfinite numbers

A parallel argument (suitably adjusted for the different context of a convergent sequence) applies to the dog that eats the least amount of food.

And actual infinity is the completed infinity. — ssu

Forgive my stupidity, but I don't understand what a completed infinity is.

No harm no foul I hope. — fishfry

OK. No problem. I was just annoyed because I didn't understand what was going on.


Thanks for your message. As you see, it's sorted out now.

Revenge? What do you mean? By writing a long post? — fishfry

Yes. I was saying in a complicated way, that a long post is not, for me, a bad thing.

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. — fishfry

That's a useful tactic. I shall use it in future.

Yes he got in trouble for harassing his female doctoral students. — 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.

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. — 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.

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. — 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.

Ok, bounds. They're just the shoulders of the road. Thing's you can't go past. Guardrails. — fishfry

That's a very helpful metaphor.

If I am understanding you, you think time is somehow sneakily inherent in math even though I deny it.
Have I got that right? — fishfry

Yes.

I cannot fathom what you might mean. — fishfry

Nor can I. That's the problem.

The subject matter of mathematics does not speak about time. — fishfry

That's the starting-point.

A sequence does not approach its limit in time. — 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.

I don't think mathematicians talk about supertasks. They're more of a computer science and philosophy thing. — fishfry

Yes. I realize this is border country. Godel seems to live there too.

Some time ago I mentioned time dilation in relativity theory in this regard. — 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.

The convergent series is misrepresented as "stopping", because the end of "stopped is never achieved. — Metaphysician Undercover

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".

We like to round things off. — Metaphysician Undercover

If you are right about relativity, I wouldn't disagree.

No. I'm complaining that things I have never said - and do not know enough mathematics to articulate - have been attributed to me.

All I know is that a link to a message that I did post, on the Infinite Staircase thread, has been added to text and quoted by fishfry in this post on the Fall of Man thread. Judging by this post and my exchanges with them on the Fall of Man thread, they did not do this. I don't know who did it and have no way of finding out. The suggestion that it came from keystone came from .

I hope that's clear. I would like the false attribution to be corrected or removed.

Right, except for the kinds of realism that make it about the physical world, but that is one type among many. — Lionino

This is not one of those cases. The world here is meant by everything that is not created by the mind (realism X anti-realism), not just what is located in space-time (physicalism). — Lionino

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?

Warning, Long-assed post ahead. Please tell me if I'm on target with your concerns. — fishfry

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.

The mathematicians takes the kettle off the stove and places it on the floor, reducing the problem to one that's already been solved. — 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.

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. — 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.

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. — 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.

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? — 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.

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. — 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".

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. — 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.

Glad it makes sense, but the limit is NOT repeat NOT part of the sequence. — fishfry

I know that. It's not a problem. If I said anything to suggest otherwise, I made a mistake. Sorry.

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 ω in the EXTENDED natural numbers — fishfry

... because "1/2, 1/4, 1/8, .." gets near and stays near 0. Yes?

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. — fishfry

I understand that distinction.

There is no time in mathematics. But supertasks are all about time. That's where a lot of the confusion comes in. — fishfry

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. — 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.

Time is not a consideration or thing in mathematics. All mathematics happens "right here and now." — 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.

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. — fishfry

The real numbers are the completion of all the sequences of rationals. That's how we conceptualize the reals. — fishfry

It's clear that the sequence does not "get near and stay near" 42. — fishfry

We can think of this as a FUNCTION that inputs a natural number 1, 2, 3, ... and outputs 1/(2 to the power of n). I'm starting from 1 rather than 0 for convenience of notation, it doesn't matter. — fishfry

If n is a number, then Sn is a number, where S is the successor function. — 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.

Does the axiom of identity mean Ludwig V = keystone ?
Just curious. :smile: — jgill

Does the axiom of identity mean Ludwig V = keystone ?
— jgill
LOL I don't think so but I see what you mean. — fishfry

I'm within epsilon. I no longer have any idea what we are conversing about. — fishfry

If you click on the link to the quotations in your message, you will find yourself here:-

I thought that might be your answer. Perhaps we shouldn't pursue the jokes, though. — Ludwig V

That is my message. It is on the "Infinite Staircase" thread, and does not include any of the passages attributed to me in your quotations. So I have no idea who wrote them.

I'm not particularly pleased by this, though I'm flattered at the extent of mathematical knowledge that is attributed to me. It is so far from the truth (as fishfry knows) that it couldn't even count as a forgery or a parody.

In a way, no harm is actually done, but it is annoying and it should at least be taken down and perhaps prevention measures implemented.

I'm hoping that @Baden is enough to draw moderator attention.

The issue here is that we really know very little about the nature of the passing of time. — Metaphysician Undercover

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.

Then the point which marks the limit, midnight or whatever never comes — 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.

I agree with this, but I'd describe it as how we apply mathematics to space and time. — 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.

Now, we add a bit of "reality". Achilles will pass the tortoise, the allotted amount of time will pass. So we see that what we take for "reality", is inconsistent with, or contradicts what the thought experiment asks us to consider. — 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.

However, "because there always has been" does not provide proof that there will continue to be into the future. — 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.

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. — 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.

If we have made a continuous uninterrupted journey from A to B we can be said to have covered all the stretches described in the first premise; that is, our motion can be analyzed as covering in turn AA', A'A", etc. (his italics) — Benacerraf on Supertasks p. 766

That's exactly right. — Metaphysician Undercover

I'm glad you agree. And you are right to go on to consider choices we could make.

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. — 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.

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. — fishfry

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?)

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, ..., ω such that ω is a limit point of the sequence, in exactly the same way that 1 is the limit of 1/2, 3/4, 7/8, ... — 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 ω in its context. In particular, there may be limitations or complications in the sense of "arbitrary" in this context.

No. 0, 1, 0, 1, ... does not have any limit at all. And we can even prove that. — fishfry

I thought so. So when the time runs out, the sequence does not? Perhaps the limit is 42.

Also, I don't think there even is a name for an arbitrary termination value for a non-convergent infinite sequence. In this case 47 is still the value of the "extended sequence" function at ω. I call it the terminal state. — 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.

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." — fishfry

I have completist tendencies. I try to resist them, but often fail.

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. — Michael

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.

Benacerraf claimed that the supertask being performed and then the lamp being on is not a contradiction. — 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.

The price is that the final state will not be reached from the previous states by a convergent sequence. But this by itself does not amount to a logical inconsistency. — SEP on Supertasks

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.

I'm sorry. There is a serious typo in The first sentence of my reply to you. It should have read:-

Not quite. If the last stage of the supertask was odd, it is not on spontaneously and without cause. — 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.

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. — Michael

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.

ω can be defined such that it is the limit of the sequence of the natural numbers. — fishfry

Quite so. Except I thought that it had actually been done.

Neither 0 nor 1 is the limit of the sequence of alternating 0's and 1's. — 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 ωth entry?

w is a limit ordinal, and it is the ordinal limit of the sequence of all the natural numbers. — 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. ω is not used in any other context - so far as I know.

What justifies such an assumption with regard to an entirely fictional lamp, coach, or pumpkin? — 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.

My charity ran out long ago regarding this subject. The lamp is a solved problem. — fishfry

Can I ask what your solution is? Just out of interest.

No, I mean they are inconsistent. To be consistent with the rules is to act according to the rules. Actions which are outside of the rules are not according to the rules, therefore they are inconsistent with the rules. — 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.

Any reasonable person should infer that nothing else happens between 10:01 and 10:02. Even though this is a physically impossible imaginary lamp, and even though I haven't told you what happens at 10:02, it is poor reasoning to respond to the question by claiming that the lamp can turn into a plate of spaghetti. The correct answer is that because 10100100 is an even number, the lamp will be off at 10:02. — 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.)

Pending completion

(Some)... compound expressions suffer the fate I attribute to 'completed infinite sequence of tasks' and 'thinking robot'. What seems most notable about such compounds is the fact that one component (e.g., 'infinite sequence') draws the conditions connected with its applicability from an area so disparate from that associated with the other components that the criteria normally employed fail to apply. We have what appears to be a conceptual mismatch. — Benacerref on Supertasks

I'm not sure whether that doesn't amount to a contradiction or whether it is an entirely distinct issue. But it seems like that if that's the case, one doesn't get as far as a contradiction.

But this implies that one has to consider what sense can be attributed to, for example, "complete" in the context of its use. It is already clear, isn't it, that the meanings of various words have to change to be applied in the context of "infinity". Just because we understand what it is to complete a finite sequence, it doesn't follow that we understand what it is to complete an infinite sequence. "Countably infinite sequence", for example, doesn't mean what you might think it means. Each and every term in the sequence can be counted, but you couldn't complete a count of all of them. In finite cases, counting each one ends up counting all. "Larger" and "smaller", if I've understood right, have to be re-defined as well.

For me, the most persuasive example is that I want to be able to say that as Achilles passes the winning post, he is completing Zeno's sequence. Or, at least, I think it is reasonable to say it. That task, of course is very different from Thompson's lamp because the sequence on/off,... doesn't have a limit, so I don't see that working in that case.

Did you mean that the phrase "completed infinite sequence of tasks" is self-contradictory? If so then yes. — Michael

You are right, of course. I'm glad you could decipher what I meant to say.

Those like Benacerraf and fishfry either claim that it isn't self-contradictory or that it hasn't been proven to be self-contradictory. — Michael

Benacerraf's position is a bit more complicated than that.

Thomson is ... successful in showing that arguments for the performability of super-tasks are invalid and ... nevertheless his own arguments against their possibility suffer the same fate. — Benacerraf on Supertasks

Those like Benacerraf and fishfry either claim that it isn't self-contradictory or that it hasn't been proven to be self-contradictory. — Michael

Thanks for clarifying that you meant self-contradictory. I've been wondering what your conclusion contradicted.

Any completed sequence of tasks is necessarily finite. — Michael

Quite so. And the phrase "completed sequence of tasks" is self-contradictory. So what do we need your argument for?

You don't get to invent your own premises and stipulate that some magical gremlin turns the lamp into a plate of spaghetti at 10:02. In doing so you are no longer addressing the thought experiment that I have presented. — Michael

Your thought experiment, your rules. But whose thought experiment is Achilles' race and Thompson's lamp? I had the impression that they are Zeno's or Thompson's. What if there's something wrong with them, such as they contradict each other or lead to a self-contradictory conclusion?

Neither is pushing the button 10100100 times within one minute, but we are still able to reason as if it were possible and deduce that the lamp would be off when we finish. That's just how thought experiments work. — Michael

True. I wrote carelessly. What deduction do you make when you think about pushing the button after an infinite sequence, which is defined without completion, of button pushes within one minute. Oh, wait, I know.

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 — Michael

You know perfectly well that's self-contradictory, so necessarily false. Ex falso quodlibet otherwise known as logical explosion. Or your deduction is wrong. (But I don't think it is wrong - or at least, not any more wrong than the spaghetti).
Not my rules. Yours.