## Two Philosophers on a beach with Viking Dogs

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I can't fathom it would be for anybody else.ssu

Maybe I asked the wrong question.
If all of the dogs are fed, is there anything left over? Until it is time to feed them again at least. Or does the food continue to be 100% even if some of it is removed?
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To show one way how an at least 2400 year old (but likely older) difficulty in mathematics emerges, which hasn't gone away. You should read the answer that I gave to L'éléphant and @javi2541997 here. It gives also a question for further thinking.ssu

Yes, I already read that, and I didn't see much to disagree with, except your question at the end.
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and @Metaphysician Undercover

You both had a very interesting exchange. I am sorry, ssu. His reply to me and Elephant was awesome, but I didn't know what to answer back because I do not have a big background in math and logic. The replies by MU are pretty good too.
Under my very basic sense of logic or math, I still root for Zeno because of the following: by randomly picking up a dog and then starting to count from it the various quantities other dogs, was missing at least these two dogs, one that ate the least and one that ate the most. After all, didn't their amounts that they ate differ from all the other dogs?

I agreed. Even if the dogs are uncountable, at least one will eat the most, followed by the least. But this is only a very basic concept of mine. I can't keep debating with logic or numbers, as you did. But, sure, I believe Zeno's two dogs must exist since there is always a "most" and a "least," correct?
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Consider this example, suppose we want to set a scale to measure all possible degrees of heat in the vast variety of things we encounter, a temperature scale. We could start by determining the highest possible temperature, and the lowest possible temperature, (analogous to Zeno's dogs) and then scale every temperature of every circumstance we encounter, as somewhere in between. Alternatively, we could start with one temperature, the freezing point of water for example (analogous to Plato's dog), and scale the temperature of all other things we encounter relative to this. Whether there is a hottest or coldest possible temperature is irrelevant to this alternative way of scaling.

Incidentally, I think this issue is relevant to the way that we judge goodness and badness in moral actions, and create codes of ethics. Some would argue that we need a best, the omnibenevolent God, and a worst, the evil devil, and all moral acts are judged in relation to these two. Others however, argue that we take any random act, and judge whether other acts are better or worse than it. I would argue that the latter is the common way that people make decisions. If a person is inclined toward a particular act (this represents "Plato's dog"), they will look at other possibilities, and judge these possibilities, each one, as to whether it is a better or worse course of action in relation to the one that the person is inclined toward. The person will choose accordingly. I believe that it is not often, that in making a decision, the person judges the possible act as to whether it is closer to what God would choose, than what the devil would choose.
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You both had a very interesting exchange. I am sorry, ssu. His reply to me and Elephant was awesome, but I didn't know what to answer back because I do not have a big background in math and logic. The replies by MU are pretty good too.
Please, I value everybody's contribution as I cannot overstate here just how difficult and open ended question this is. Yet it's very simple and you can think about it even without a long background in math. That's the real beauty of math, at it's most beautiful, it's elegant and simple.

But, sure, I believe Zeno's two dogs must exist since there is always a "most" and a "least," correct?
I agree too, wholeheartedly. But notice how radical (or outrageous to some) our view is, actually. Plato's rejection is totally logical. And think just where we come with our own thinking. If the other of Zeno's dog more than any other dog, there cannot be a dog or a collection of dogs that eat more, right? It absolutely eats more than any dog, I would boldly argue.

I'll try to show just how problematic this is even with Plato's dog and the multiples of this dog.

Let's start Plato's dog, dog1 and all those dogs that eat exactly some multiple times it's food (dog1, dog2, dog3, dog4, dog5, and so on). Let's pick three dog from this collection of dogs (or set of dogs) and have dog a, dog b and dog c that

dog a + dog b = dog c

Now if we know two dogs, we can compute the third one in the equation. So if dog a is actually dog2 and dog b is dog3, then you can come to the conclusion that dog c is of course, dog5. We can solve the equation. However, if we have an inequation like:

dog a + dog b < dog c

We don't know what dog c is exactly, even if we would know that the others (a and b) are dog2 and dog3. The only thing we can say then is that dog c can then be dog6, dog7, dog8 or a dog that eats a higher multiple than that of dog1's food. And that's it. We cannot calculate what dog c is. Dog c obviously exists (as it belongs to this set of dogs and if it's dog6 or higher) as the multiples of dog1 go on and on and never stop.

Just how confusing this becomes is when we notice that actually our definitions of Zeno's dogs are inequations:

Zeno's least eating dog eats < any other dog there exists eats
Zeno's most eating dog eats > any other dog there exists eats.

Yet we can intuitively think that Zeno's dogs exist and we have a place for them. We can assume a well ordering using the amount of tood the dogs eat as did Plato ( dog1 < dog2 < dog3 ). Yet consider then putting Zeno's dogs on each ends of the lines. What happens? You cannot pick any dogs between them. You have lost all ways of measurement. Or in other words, you cannot pick the next dog from Zeno's least eating dog or the previous dog before Zeno's dog that eats more than everybody.

And then, if you think that there's just two Zeno's dogs, how about then all the transcendental dogs between them.
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Consider this example, suppose we want to set a scale to measure all possible degrees of heat in the vast variety of things we encounter, a temperature scale. We could start by determining the highest possible temperature, and the lowest possible temperature, (analogous to Zeno's dogs) and then scale every temperature of every circumstance we encounter, as somewhere in between.
Err, isn't there actually an absolute lowest temperature, - 273,15 Celsius? We cannot talk then about a temperature of - 2 000 000 Celsius or lower temperatures to my knowledge. So this isn't similar to the problematics of the Zeno's dogs in the story (or at least the other one).
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Interesting example, friend. I now understand this tricky game a bit more. But I still don’t understand why you state that ‘Whether there is a hottest or coldest possible temperature is irrelevant to this alternative way of scaling.’ Are you suggesting that it is irrelevant to Plato whether there is a dog who eats the most and another who eats the least? Well, maybe. But the rules stated by Athena say: ‘All the dogs eat the same food, which is divisible, and there is enough of it for every dog’ but Zeno argues (and I agree with that) that by randomly picking up a dog and then starting to count from it the various quantities other dogs, was missing at least these two dogs, one that ate the least and one that ate the most.

And the second rule states that there are no constraints on quantity (physical or otherwise), and hence on dogs. So Zeno is right here. There will always be one dog who eats the most and another who eats the least, which I believe is relevant to this issue, and Plato overlooked these two dogs in his counting.
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I wholeheartedly agree with everything you wrote in your post. It is funny and interesting to debate about this, and I appreciate your effort to explain it to me.

And then, if you think that there's just two Zeno's dogs, how about then all the transcendental dogs between them.ssu

Exactly. As you mentioned, the rest of the dogs are simply transcendental in the situation. What I know for certain is that there will be at least two dogs: one that eats the most (let's call him dog >) and one who eats the least (let's call him dog <), but I'm not sure who dog b is, because the latter is just transcendental to the scale. What I can't do, if I understand Zeno correctly, is start counting by dog b or another random dog "x" because my numbering will be irregular due to forgetting those two dogs. The one who "starts" and the one who "ends," or, to put it another way, the one at the bottom and the other at the top.
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Err, isn't there actually an absolute lowest temperature, - 273,15 Celsius? We cannot talk then about a temperature of - 2 000 000 Celsius or lower temperatures to my knowledge. So this isn't similar to the problematics of the Zeno's dogs in the story (or at least the other one).ssu

That is the lowest temperature realizable from our methods of measurement. In other words it is a restriction created by our choice of dog to use for comparison, the movement of atoms. It does not mean that a lower temperature will not be discovered, if we devise a different measurement technique. Notice there is no such limit to the hottest possible temperature, because we move to different measuring principles.

Are you suggesting that it is irrelevant to Plato whether there is a dog who eats the most and another who eats the least? Well, maybe.

That is exactly what I am suggesting. Plato was given the task of measurement, and he took that task and proceeded. That the task will never be completed because the quantities are unlimited, is irrelevant. Therefore whether or not there is a dog that eats the least or the most, is also irrelevant.

But the rules stated by Athena say: ‘All the dogs eat the same food, which is divisible, and there is enough of it for every dog’ but Zeno argues (and I agree with that) that by randomly picking up a dog and then starting to count from it the various quantities other dogs, was missing at least these two dogs, one that ate the least and one that ate the most.

The "other two dogs" referred to by Zeno is a sophistic ruse, just like Plato says. Zeno could have said, "let me know when you get to the dog that eats the most, and the dog that eats the least", and Plato could have said "OK". Problem resolved. Instead, Zeno said you are "forgetting" these two dogs. But Plato is not "forgetting" them, he has not yet found them, so there is no need for them to have ever entered his mind.
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That is the lowest temperature realizable from our methods of measurement. In other words it is a restriction created by our choice of dog to use for comparison, the movement of atoms. It does not mean that a lower temperature will not be discovered, if we devise a different measurement technique.
And I thought in my ignorance, that there's at least this obvious limit in Physics! Of course, what is Physics else than the study of change and movement? So there's big problems to get funding for a research on the effects of temperatures of negative millions of Celsius. Fortunately there's an actual reality to seek something else.

That is exactly what I am suggesting. Plato was given the task of measurement, and he took that task and proceeded.
Even if this was for javi, here's my point: That wasn't the task. The task was to feed all the dogs. Plato tries desperately to please his goddesses by taking a dog as the measurement stick (dog?) and tries to get some order to the dogs. Will he accept even irrational dogs, I don't know. But transcendental dogs surely are something he didn't know and the reals are the problem. But they are should I say in the realm of being Zeno's dogs.

The "other two dogs" referred to by Zeno is a sophistic ruse, just like Plato says. Zeno could have said, "let me know when you get to the dog that eats the most, and the dog that eats the least", and Plato could have said "OK". Problem resolved.
I have to point out this: Zeno understood Plato's argument. Indeed you cannot reach Zeno's dogs from Plato's dog because of Plato's argument. It is quite valid. Or to put this in another way, the whole definition of Zeno's dogs relies on that they cannot be reached by measurement (or counting).

In fact your own argument that absolute zero being only a measurement problem is somewhat similar here, it's a limit for modern measurement as atoms cease to move. Yet if we define Physics to be only "atoms moving", then there's a categorical denial of your idea of lower than absolute zero temperatures. Luckily Physicists understand that they are only making models and theories and these that we hold now to be true can be proven wrong and new better models can be invented.
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Zeno could have said, "let me know when you get to the dog that eats the most, and the dog that eats the least", and Plato could have said "OK". Problem resolved. Instead, Zeno said you are "forgetting" these two dogs. But Plato is not "forgetting" them, he has not yet found them, so there is no need for them to have ever entered his mind.

Got it. Zeno completely comprehended Plato's reasoning, although he did not convey the correct response. Instead, Zeno assumed that Plato had forgotten two elementary dogs, which is incorrect. Plato merely dismissed them as irrelevant to his argument. However, those two dogs, the one that eats the most and the other who eats the least, exist for both Plato and Zeno. Right? :smile:
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And I thought in my ignorance, that there's at least this obvious limit in Physics! Of course, what is Physics else than the study of change and movement? So there's big problems to get funding for a research on the effects of temperatures of negative millions of Celsius. Fortunately there's an actual reality to seek something else.ssu

I've read some speculations showing that the hottest temperature will actually end up being the same as the coldest temperature. Strange.

The task was to feed all the dogs.ssu

If that's the case then both Plato's dog and Zeno's dogs are irrelevant, all one needs to do is point the dogs to the food and tell them to go to it. When they all get fed the task is complete. Since it is stipulated that the quantities are unlimited, the task will never be completed, some dogs will not finish eating before Plato and Zeno pass on.

However, those two dogs, the one that eats the most and the other who eats the least, exist for both Plato and Zeno. Right? :smile:

Not under the assumption that quantities are unlimited.
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If that's the case then both Plato's dog and Zeno's dogs are irrelevant, all one needs to do is point the dogs to the food and tell them to go to it.
That's why the task was for the philosophers "to tell a way to feed all the dogs on the beach without any dog being left out hungry and Themis would make this instantly to happen".
If the task is to give a goddess a way to "sort them out", then it's not a reply to have "the gods sort them out". Remember if there is an endless amount of food, there is also an endless number of dogs.
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Zeno completely comprehended Plato's reasoning, although he did not convey the correct response. Instead, Zeno assumed that Plato had forgotten two elementary dogs, which is incorrect. Plato merely dismissed them as irrelevant to his argument. However, those two dogs, the one that eats the most and the other who eats the least, exist for both Plato and Zeno. Right? :smile:

Not under the assumption that quantities are unlimited.
Plato doesn't accept the existence of Zeno's dogs. Or in reality, Aristotle and many in the following Centuries believe that there is only a potential infinity, not an actual infinity. Many finitists still this day don't believe in actual infinity, perhaps any infinity altogether. And Absolute Infinity is even more controversial.

Maybe I asked the wrong question.
If all of the dogs are fed, is there anything left over? Until it is time to feed them again at least. Or does the food continue to be 100% even if some of it is removed?
There doesn't have to be any surplus, as this is done once. The task is that the philosopher is to define in some way all the amounts of food and hence all the dogs, that they don't leave some dogs out. As no dog eats the same amount, then it's easy for the goddes to put the dogs in an growing or decreasing line based on their amount of food.
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That's why the task was for the philosophers "to tell a way to feed all the dogs on the beach without any dog being left out hungry and Themis would make this instantly to happen".ssu

Then why isn't Plato's way the proper way? There's no need to determine the dog which eats the most or the dog which eats the least, just keep feeding in the way Plato described.
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There doesn't have to be any surplus, as this is done once. The task is that the philosopher is to define in some way all the amounts of food and hence all the dogs, that they don't leave some dogs out. As no dog eats the same amount, then it's easy for the goddes to put the dogs in an growing or decreasing line based on their amount of food.ssu

Ok, so if there is no surplus and no dog that goes without food then there has to be one that will eat more than the rest and one that will eat less.

The only other option would be an infinite amount of food and an infinite number of dogs. And as you said there is a limited area in Greece, so that does not seem likely.

And even if there were an infinite number of dogs, there would still have to be the one that eats less than any other dog.
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Plato doesn't accept the existence of Zeno's dogs.ssu
and @Metaphysician Undercover @Sir2u

I guess that’s why he [Plato] explained again to Zeno that there cannot be a dog that eats the most, because there is always a dog that eats more. I am still confused regarding Plato’s argument. Yes, it is clear to me that Plato roots for infinity counting of dogs, but I think he forgets two basic dogs: the one who eats the most (the dog at the peak) and the dog who eats the least (the dog at the bottom). I think the argument of Zeno is more plausible. On the other hand, Plato argues that there cannot be a dog that eats the most, because there is always a dog that eats more. He sees infinity towards the maximum. But what about the dog who eats the least? If there is always a dog that could eat less, there will be a dog who will eat nothing at all. How can it be possible to find a dog who will eat less than previous dogs and so on? I think this has to be switched and follow Zeno’s point of limited counting: Zeno's least eating dog eats < any other dog there exists eats, and then start to count. Agree?
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On the other hand, Plato argues that there cannot be a dog that eats the most, because there is always a dog that eats more.

Does infinity actually mean that there is always one more, or does it just mean the possibility of it?
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Does infinity actually mean that there is always one more, or does it just mean the possibility of it?

I think Plato means that there is always one more dog. If we take this to maximum, it could be, somehow, plausible, and I guess I have to agree with Plato.
But I cannot see it in regression. It is not plausible to think that there is always one dog who eats less than the previous one. Yes, I know that Athena stated there was enough food for every dog. But my point is that, sooner or later, we will reach the bottom and there will finally be a dog who eats less than all the “infinity” intermediate dogs. I think this is more plausible than to think that there will always be one dog and another in both extremes continuously in a loop.
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Then why isn't Plato's way the proper way? There's no need to determine the dog which eats the most or the dog which eats the least, just keep feeding in the way Plato described.
Well, if it's so, then the counterarguments of the actual Zeno of Elea gave us are quite relevant.

And if you think that is nonsense, how about then the idea of the infinitesimal? Obviously something that created a huge debate at the time of Newton and Leibniz. The idea of an infinitesimal comes closest to the other of Zeno's dogs in the story. Remember that there's Robinsons non-standard analysis. Here's from Wolfram Mathworld:

Nonstandard analysis is a branch of mathematical logic which introduces hyperreal numbers to allow for the existence of "genuine infinitesimals," which are numbers that are less than 1/2, 1/3, 1/4, 1/5, ..., but greater than 0. Abraham Robinson developed nonstandard analysis in the 1960s. The theory has since been investigated for its own sake and has been applied in areas such as Banach spaces, differential equations, probability theory, mathematical economics, and mathematical physics.
It sure sounds a lot like the other Zeno's dog, doesn't it? And why is then non-standard? Well, basically because of Aristoteles and his following (or Plato in the story).

And how about then calculus or mathematical analysis in general? It's very useful, an important area of mathematics. But can you put it on a sound footing just with assuming Plato's potential infinity? Some argue, and in my view convincingly, that set theory was intended to put finally analysis on a firm footing by set theory. But then set theory itself stumbled into paradoxes.

The point of the story is that this problem hasn't been solved. And it comes down to the problem in the story.
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Does infinity actually mean that there is always one more, or does it just mean the possibility of it?
If it would only be possible that there could be a dog, but there wouldn't be that next dog, then obviously the number of dogs on the beach would be finite.

So potential infinity means that there is always more eating dogs ...and less eating dogs, that this process doesn't stop. Thus there cannot be the dog that eats the most or the least. This is in the story Plato's argument.

And actual infinity is the completed infinity. In the story it's basically the more eating Zeno's dog. Think about it this way: All the dogs eat something. If all they eat something, doesn't this the mean there exist the amount of food that all the dogs eat? If so, by rule #2, then there's a dog that eats it. That in the story is Zeno's argument.

(And again I tip my hat to the reasoning that L'èléphant gave on page 1.)
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The point of the story is that this problem hasn't been solved. And it comes down to the problem in the story.ssu

That's one problem. I'm sure there's many more. Find another, and start another thread.
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Not only is it one problem, I think it's been the largest problem there has been in mathematics. Just look at the long historical debate around the mathematics of continuous change and simply the history of Analysis. Yes, we use infinity as a limit point in calculus and Zeno's paradoxes are solvable by modern calculus, yet the philosophical reasoning remains open. People wanted for set theory to be the basis of mathematics as it would have given a foundation to analysis.

And furthermore, I think that today we might be closer to a solution on these open questions because we are already comfortable of there being the non-computable and non-provable but true mathematical statements. This is actually a real sea change from the time when the paradoxes of set theory were found over hundred years ago or what people thought earlier. The existence of non-computable and even non-provable mathematics would have been quite a heresy in earlier times, but now we start to accept this. (See for example another current PF thread talk about this and about Noson Yanofsky's paper "True but unprovable" here.)

The non-computability of Zeno's dogs in the story should be (hopefully) obvious. But this non-computability goes a lot more further. Set theory shows this well and the problems that naive set theory had even more.
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Rather, by rule #2, the one that eats "the most" and the one that eats "the least" are conceptual quantities that differ from any other quantities already given. — L'éléphant

Yes. But there is the supposition that how much they eat can change. To establish individuation, you need an additional criterion that is not empirical.
But you're missing the point, I think. We don't know when they stop counting of how much each dog eats -- whether going up or downwards quantity. They could continue counting, for all I care. But the fact remains that there is the dog the eats the most and the dog that eats the least. Plato and Athena would not know this until after they stop counting (that is, if they could stop counting). But already Zeno identified two dogs that eat differently than their dogs.
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But already Zeno identified two dogs that eat differently than their dogs.
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.
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.
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Plato and Athena would not know this until after they stop counting (that is, if they could stop counting).
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.

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.
I think everybody understands that there is no largest finite number. Because, every natural number is finite, right? Even in the story Zeno is well aware of this.

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.
(First of all, notice that ω here refers to the largest Ordinal number. In the story it would mean that you put all the dogs that food amount is exactly divisible by dog 1's food (let's call them positive dogs) in a line from smaller to bigger, and then 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. Notice it's different from cardinal numbers.)

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. Why then couldn't it be a dog on the beach?

After all, limit sequences are the way we also defined the other of Zeno's dogs. Yes, we refer to limits and only non-standard analysis to infinitesimals, however the modern calculus does go the lines of Leibniz, who used the infinitesimal, which is the least eating Zeno's dog in the story:

Modern derivative and integral symbols are derived from Leibniz’s d for difference and ∫ for sum. He applied these operations to variables and functions in a calculus of infinitesimals. When applied to a variable x, the difference operator d produces dx, an infinitesimal increase in x that is somehow as small as desired without ever quite being zero. Corresponding to this infinitesimal increase, a function f(x) experiences an increase df = f′dx, which Leibniz regarded as the difference between values of the function f at two values of x a distance of dx apart. Thus, the derivative f′ = df/dx was a quotient of infinitesimals.

Forgive my stupidity, but I don't understand what a completed infinity is.
Well, you already referred to completed infinity or actual infinity with the example of ω as that is Cantorian set theory. Here's one primer about the subject: Potential versus Completed Infinity: its history and controversy
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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 biggerssu
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.
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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

:up:
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A transfinite number isn't a natural number, so it doesn't get attached to (aligned with) a dog. Nor could it be.
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? And what about transcendental dogs? They are finite, but the dog that eats π amount compared to Plato's dog?

(And here I have to make a correction to above. As all dogs do eat something, we have a problem with the non-existent dog that doesn't eat anything, as that is part of the natural number (natural dogs) and I should have referred to positive integers (positive dogs, not natural dogs).

That will take you, and even the gods, an infinite time.
Now your are putting physical limitations to the story, which didn't have them (Athena created the dogs instantly and Themis could feed them instantly also, if given the proper rule / algorithm). In fact when you think of it, already large finite number of dogs cause huge problems in the physical world: if counting or feeding a dog takes even a nanosecond, with just finite amounts of dogs the whole time universe exists won't give enough time to count or feed them. If your counterargument is ultrafinitism, that's totally OK. This is a Philosophy Forum and this issue is totally fitting for a philosophical debate. I would just argue that the system of counting that basically is like 1,2,3,4,...., n, meaningless over this number isn't rigorous. It's very logical to have infinities as mathematics is abstract.

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.
Well, I gave you already on article going over this earlier. Just a quote from it, if you don't have the time to read it:

Potential infinity refers to a procedure that gets closer and closer to, but never quite reaches, an infinite end. For instance, the sequence of numbers
1, 2, 3, 4, ...
gets higher and higher, but it has no end; it never gets to infinity.

Completed infinity, or actual infinity, is an infinity that one actually reaches; the process is already done. For instance, let's put braces around that sequence mentioned earlier:
{ 1, 2, 3, 4, ... }
With this notation, we are indicating the set of all positive integers. This is just one object, a set. But that set has infinitely many members. By that I don't mean that it has a large finite number of members and it keeps getting more members. Rather, I mean that it already has infinitely many members. We can also indicate the completed infinity geometrically.
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As all dogs do eat something, we have a problem with the non-existent dog that doesn't eat anything,ssu
and @Ludwig V

I remember I referred to that specific dog in our previous exchange. I said that following Athena’s rule, it is not possible to think that there will always be a dog that will eat less than the previous one, and so on. Athena stated that there is enough food for every dog. 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. Those are the two dogs that Zeno was referring to: the dog at the “top” and the dog at the "bottom,” but why do you count a non-existent dog? If there is enough food for the dogs, there isn't a dog who doesn’t eat anything at all.
I mean, following the premises of the OP it is not possible to imagine a dog who doesn’t eat anything.
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