It all comes down to rule2 and how we interpret rule1. By rule2 if there is an amount, there's a dog for it. If nothing is an amount, then there is a dog for that. Now if rule1 eating means that a dog cannot refrain from eating, then obviously it's a non-existing dog with a non-existing amount of food. Now if we want to include that in the or not is in my view a philosophical choice (and in reality it took a lot of time for Western mathematics to accept zero as a number).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. — javi2541997
I don't recall mentioning any non-existent dogs, nor any that don't eat anything.As all dogs do eat something, we have a problem with the non-existent dog that doesn't eat anything, — ssu
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
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.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
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.And what about transcendental dogs? They are finite, but the dog that eats π amount compared to Plato's dog? — 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.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
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
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. — Ludwig V
Notice that π isn't constructible, but the square root of two is if irrational, is not transcendental.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. — Ludwig V
Your ordering means you have to start from a dog that you cannot identify. — Ludwig V
The transcendental food was a joke, playing on the absurdity of transcendental dogs. I must be more careful about jokes.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
That is only possible if there is a finite number of 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
Well, strictly speaking they are identified by the amount of food they eat, which determines their position in the line.Honestly, I think those two are always ‘there’ but it is a mistake to try to identify them with numbers. — javi2541997
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. — Ludwig V
Well, strictly speaking they are identified by the amount of food they eat, which determines their position in the line. — Ludwig V
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. — Ludwig V
Is there a non-existing dog? If there is, it doesn't exist. If there isn't, it doesn't exist.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
Exactly.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
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.Sorry, I was foolish in trying to follow usual norms when infinity is involved. :sweat: — javi2541997
Is there a non-existing dog? If there is, it doesn't exist. If there isn't, it doesn't exist. — Ludwig V
That's a complicated thought process. This is a story. It was made up. Speculations about what Athena thought or didn't think beyond what we are told in the text can be plausible or implausible but there's no criterion for truth or falsity. The same applies to ideas about what Plato would or would not have done. For what it's worth, I don't think the real Plato would have done any of what the story attributes to him. But it doesn't matter. But there's no truth or falsity beyond what is stated in the the text - and what follows logically from that.I think Athena never thought about it either. But since this mysterious dog showed up in this game yesterday, I started to think about his interference in the counting. Well, if we imagine there is actually a dog who doesn’t eat anything, it means that it should be represented with a zero (0) in the counting. As ssu pointed out, it took a while for Western mathematics to accept zero as a number. According to this issue, maybe Plato would never have taken the dog who doesn’t eat anything into account, but yet it is clear we should take the dog into account, and thus, the dog exists. Right? — javi2541997
I'm sorry to be a bit abrupt, but if you don't keep your feet on the ground, you're bound to lose contact with reality. — Ludwig V
Actually not.It is true that my knowledge of mathematics and logic is pretty limited. Yet, if I understand the rules of this entertaining game correctly, the counting starts with two identified 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
Bravo.Honestly, I think those two are always ‘there’ but it is a mistake to try to identify them with numbers. — javi2541997
I don't think you have derailed anything. If there's any derailing going on, it's me that's doing it.I am the one who apologises for derailing the topic in an inconsistent scenario — javi2541997
You can do that, but it's very misleading. It suggests that an infinite line is just a very long line. That's wrong. The best way I can think of is to draw your line and put your ∞ or ω at the end of it, but remember that those symbols mean that the line goes on forever - it has no end. That's why we always just write down the first few elements of the sequence and then ... or "and so on". That's not just an abbreviation or laziness or lack of time. It's telling you that the sequence has no end.Just think of a finite line you draw and put at the start zero and in the end — ssu
I don't know about all those theorems. I know I should, but I had a deprived education.The whole story is about the problem of definition that math has. And for the Grand Order you refer to, there is the Well Ordering Theorem. — ssu
Well, we can talk about the set of all natural numbers ℕ, right? I don't think that it's misleading.You can do that, but it's very misleading. It suggests that an infinite line is just a very long line. That's wrong. — Ludwig V
That's exactly what I have been trying to say all along! :smile: — Ludwig V
I see your point.And here's then the problem: not only Plato started from counting, but even today Set Theory starts from counting too with the Peano Arithmetic. — https://en.wikipedia.org/wiki/Class_(set_theory)ssu
Yes. I always thought that was the point. Why should everything have a definite, computable result? Stating the range of a result is not pointless.Can you know or compute C, if you know both A and B? No, if A and B are as above, then only thing you know is that C can be a natural number 6 or 7 or 8 or larger. It might be six, but then it might be three googol also. — ssu
I think it's good to go this through here. So the basic problem was that "Naive Set Theory" of Frege had this Basic Law V, an axiom schema of unrestricted comprehension, which stated that:I don't quite get that "fork" argument. The notation using lower case beta for a member of the set and upper case beta for the set is confusing, and I think there's a typo in the statement of the paradox. But I know better than to challenge an accepted mathematical result. — Ludwig V
For any two concepts it is true that their respective value ranges are identical if and only if
their applications to any objects are equivalent.
Unfortunately... yes.That's always a good solution to a difficulty - slap a name on it and keep moving forward. Sometimes mathematicians remind me of lawyers. — Ludwig V
Believe it or not, I can see that.This simply goes back to in the story of Plato's rejection of Zeno's most eating dog, just in a different form. — ssu
I'm a bit confused about infinitesimals. Are they infinitely small? Does that mean that each one is equal to 0 i.e. is dimensionless? Is that why they can't be used in calculations? (I thought that Newton used them in calculus and Leibniz took exception.)This is why idea of infinitesimals is rejected in standard analysis. — ssu
Well, actually, someone else mentioned it. I misunderstood what it is about and off we go. Once I realized it was about the sum of an infinite sequence, I withdrew, with some embarrassment. But I've learnt some interesting snippets.In fact you yourself brought up an old thread of four years ago, which is topic sometimes even banned in the net as it can permeate a nonsensical discussion. — ssu
There is another way, mentioned in the video. Just relax and live with your paradox. It's like a swamp. You don't have to drain it. You can map it and avoid it. Perhaps I just lack the basic understanding of logic.That's always a good solution to a difficulty - slap a name on it and keep moving forward. Sometimes mathematicians remind me of lawyers.
— Ludwig V
Unfortunately... yes. — ssu
:grin:Believe it or not, I can see that. — Ludwig V
Both Newton and Leibniz figured out the way to make a derivation by using infinitesimals.I'm a bit confused about infinitesimals. Are they infinitely small? Does that mean that each one is equal to 0 i.e. is dimensionless? Is that why they can't be used in calculations? — Ludwig V
Well, in my view mathematics is elegant and beautiful. And it should be logical and at least consistent. If you have paradoxes, then likely your starting premises or axioms are wrong. Now a perfect candidate just what is the mistake we do is that we start from counting numbers and assume that everything in the logical system derives from this.There is another way, mentioned in the video. Just relax and live with your paradox. It's like a swamp. You don't have to drain it. You can map it and avoid it. Perhaps I just lack the basic understanding of logic. — Ludwig V
I don't get this. There's enough food for all the dogs, so why does it have to take some from Plato's dog? If it does, then of course the amount of food for Plato's dog has decreased, but the food supply is infinite, so the amount of food available overall hasn't decreased. What's the problem?Zeno's least eating dog has to eat something, but then if let's say eats from Platons dog 1, then the food hasn't decreased! — ssu
Right from the beginning, 2,500 years ago, people have been thinking that everything has been done and is perfect. But then they found the irrationality of sqrt(2) and pi. A paradox is not necessarily just a problem. Perhaps It's an opportunity. Oh dear, what a cliche!Well, in my view mathematics is elegant and beautiful. And it should be logical and at least consistent. If you have paradoxes, then likely your starting premises or axioms are wrong. Now a perfect candidate just what is the mistake we do is that we start from counting numbers and assume that everything in the logical system derives from this.
And if someone says that everything has been done, that everything in ZFC works and it is perfect, I think we might have something more to know about the foundations of mathematics than we know today. — ssu
Writing x^2 means x². A bit lazy to use this way of writing the equation.I'm afraid I don't know what "^" means. — Ludwig V
Exactly. With limits we want to avoid this trouble. Yet it isn't actually a paradox as infinitesimals are rigorous in non-standard analysis.But the paradox in the concept of the infinitesimal - that it both is and is not equal to zero - Is not difficult to grasp - and I realize that that's what the concept of limits is about. — Ludwig V
It doesn't. This isn't part of the story, I just wanted to describe the seemingly paradoxical nature of the infinitesimals. And hence when infinitesimals had this kind of attributes, it's no wonder that bishop Berkeley made his famous criticism about Newtons o increments (his version of infinitesimals):I don't get this. There's enough food for all the dogs, so why does it have to take some from Plato's dog? — Ludwig V
“They are neither finite Quantities nor Quantities infinitely small, nor yet nothing. May we not call them the Ghosts of departed Quantities?
I agree. Perhaps they admit that there's just only some minor details missing, that aren't so important.Right from the beginning, 2,500 years ago, people have been thinking that everything has been done and is perfect. — Ludwig V
I think it's already satisfying to know just what issues we don't know, but possibly in the future could know. And I think there's still lot to understand even from the present theorems we have.But then they found the irrationality of sqrt(2) and pi. A paradox is not necessarily just a problem. Perhaps It's an opportunity. Oh dear, what a cliche! — Ludwig V
Thanks. One has to do something when one doesn't have the keyboard for the symbolism. Handwriting is much more flexible.Writing x^2 means x². A bit lazy to use this way of writing the equation. — ssu
I didn't realize that argument was so powerful.Yet using the diagonalization method we get also many other very interesting theorems and proofs and also paradoxes, which in my opinion are no accident. — ssu
He was a great wit. I'm still trying to make up my mind whether he was a great philosopher or a complete charlatan - even possibly both. This comment is typical. It is very sharp, very pointed. But the calculus is embedded in our science and technology.“They are neither finite Quantities nor Quantities infinitely small, nor yet nothing. May we not call them the Ghosts of departed Quantities?
Yes, I see. You can remove an infinitesimal amount from a finite amount, and it doesn't make any difference - or does it?I just wanted to describe the seemingly paradoxical nature of the infinitesimals. — ssu
What do you mean by "actual infinity"?Set theory gives us the actual infinity — ssu
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.
Calculus or analysis is the perfect example of us getting the math right without any concrete foundational reasoning just why it is so. Hence the drive for set theory to be the foundations for mathematics was basically to find the logic behind analysis.This comment is typical. It is very sharp, very pointed. But the calculus is embedded in our science and technology. — Ludwig V
To my reasoning it doesn't. And both Leibniz and Newton could simply discard them too with similar logic.Yes, I see. You can remove an infinitesimal amount from a finite amount, and it doesn't make any difference - or does it? — Ludwig V
I'll give the definition from earlier:What do you mean by "actual infinity"? — Ludwig V
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.
Oh dear! That's a real can of worms, isn't it? Some philosophers would argue that the engineers have got it right. Perhaps it is best to start with the foundation of philosophy - a question. "What do you mean by a foundation?" But I do know that some mathematicians regard philosophers in much the same light as they regard engineers. Still, it's all great fun and often elegant and beautiful; I don't want t be a grinch.Calculus or analysis is the perfect example of us getting the math right without any concrete foundational reasoning just why it is so. — ssu
Yes. I remember. I don't think I ever replied properly. I can see why those definitions might seem reasonable. But it seems better to me to say that "potential", "actual" and "complete" have no application here. On the other hand, I can see that there are real problems here, so I'm not sure that these labels matter very much. Do they solve any problems?All of these sets are of finished "actual infinity", not the potential infinity as the Greeks thought. — ssu
The trouble is that, like plastic, if you discard them, they just come back to haunt you. Perhaps Berkeley had a point. Perhaps the concept of incommensurability could help here?To my reasoning it doesn't. And both Leibniz and Newton could simply discard them too with similar logic. — ssu
Definitely.Perhaps Berkeley had a point. Perhaps the concept of incommensurability could help here? — Ludwig V
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