Plato and Athena would not know this until after they stop counting (that is, if they could stop counting). — L'éléphant

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. — Ludwig V

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. — Ludwig V

(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. — Ludwig V

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

I live in Russia (please note that I support Ukraine). — Linkey

If so, please be careful @Linkey. And welcome to the Forum.

I am sure that Russians will vote in this referendum to end the war. If the war continues, Russian soldiers will be unable to fight, because they will suffer from cognitive dissonance - what are they fighting for? For censorship and repression? — Linkey

All the Russian emigrants living in my country that I've spoken to don't like what Putin did by attacking Ukraine, many were simply horrified, but then again they don't live Russia. Only once have I seen in 2014 in Helsinki two young Russian men openly in public wearing the black orange stripes of the ribbon of Saint George. Yet 2014 isn't 2022 or today.

Yet I think there are still Russians who support the war simply fearing what will happen to Russia if the war is lost. You see, Russia isn't a normal nation-state, it still is built on an Empire. That's the real problem. Still many Russians believe Catherine the Great's words: "I have no way to defend my borders but to extend them." This pure imperialism hasn't yet died in your country.

And the worst thing is that now other countries simply won't trust Russia. You did totally surpise the West with the collapse of the Soviet Union, but it was Russia itself wanting the destruction of the Empire.

Assume if Putin's regime falls and new not so hostile towards the West administration takes over. Well, a lot of people in the West won't believe that this administration will continue to hold firmly power and assume that we can in the West might (again) wake up with coup again in Moscow and a new regime that builds statues for Putin the Great and declares to the Russian people how evil the West is and how it's real intention is to destroy Russia.

I hope I will not violate the forum rules, if I propose the easiest way for the West to defeat Putin and Xi. First, the United States should reconsider its nuclear doctrine, and declare that the use of US nuclear weapons is possible only in the form of a symmetrical response. If Putin nukes one city, the United States would nuke one Russian city, if Putin nukes ten, the United States would nike ten, and so on. — Linkey

With nuclear weapons there's always strategic ambiguity: you won't really tell what you're response is and even if you tell it, it's likely that others won't believe you. And you don't want to tie your hands. Now it is likely that a nuclear exchange might well become a tit-for-tat, isn't at all sure that nuclear war would go this way. Once you have crossed the line and have used nukes, it's a whole new World: use of nuclear weapons is normal. People will adapt to it.

how are you supposed to be a part of the same "demos" with these (distant to you) people? How is democracy supposed to work in such a scenario (that seems very plausible in many developed countries)? — Eros1982

Before going further, Let's remember first that democracy is a system of government and a state or a country is a different thing. Even if the OP doesn't take this into account, I think it is very important to understand that "people not feeling part" of a country is a very alarming issue for any state, be it democratic or not.

First and foremost the "demos", meaning the people, is inherently important for any state or country to exist independent of the system of government. The people that make the inhabitants of the state have to share an idea about their state. This is why for Empires and states that have in themselves clearly separate people with separate languages and cultures, even religions, have structural problems today. And even quite established democracies like the United Kingdom or Spain can have secessionist movements. Empires like Russia and China have obvious problems and have resorted to what some can rightly call genocidal actions (Russians with the Chechen's and China with the Uighurs).

This wasn't what the OP had in mind, but I think it's very important to understand this aspect before answering further the OP.

Now we come with the question what happens in countries where there are no dominant cultures and apart from abiding to state laws, no traditions and no values are taken to be the norm. — Eros1982

This is something that is argued to happen especially if what is promoted is "multiculturalism". And that multiculturalism destroys the norms, traditions and the values.

It would be good to observe first how actually norms and traditions change before talking about their destruction. Because I would make the claim there indeed still are norms and even traditions.

Then the question about "no dominant culture". Well, our global culture has morphed into something quite similar to a dominant culture. We read the same books, listen to the same music, look at the same films. How our own "nation state culture" survives in the Global village is a difficult question. And this isn't about just the system of governance either. I would argue that this globalization and this melting pot of cultures is the real force behind how the specific culture that a specific people falls from a dominant position it perhaps enjoyed earlier.

And then you have nations and civilizations which at a point do not know anymore what they want (apart from economic growth). Who do you think will prevail? The crazy theocratists who have some definite goals or the moderate guys whose only daily dilemma is to live a pleasant life (only) or to suicide? — Eros1982

A democracy following it's will of it's people will look quite clueless about what they want simply because the people will have different opinions and goals. And this is what always should be remembered about democracies: they appear far weaker than they are.

On the other hand, totalitarian systems look far more stronger than they actually are. The collapse of the Soviet Union is the best example of this. Never had an empire collapsed due to the bankruptcy of it's ideology as peacefully and rapidbly as the Marxist-Leninist experiment did. Yet unfortunately the "normal" way how Empires fall through war and blood is now played in the war between Russia and Ukraine, something that the last Soviet leadership was able to dodge and what the current revanchist Kremlin wanted to do.

That sounds like the "New Math" they had when I was in school. I loved it but it was a failure in general.

I don't think they teach basic arithmetic anymore. It's a problem in fact. — fishfry

There's many things they don't teach in school when looking at what my children have to study. Usually the worst thing is when the writers of school books are too "ambitious" and want to bring in far more to the study than the necessities that ought to be understood.

Here's the general theorem in the setting of category theory. It's called Lawvere's fixed point theorem. Not necessary to understand it, just handy to know that all these diagonal-type arguments have a common abstract form. — fishfry

I looked at this. Too bad that William Lawvere passed away last year. Actually, there's a more understandable paper of this for those who aren't well informed about category theory. And it's a paper of the same author mentioned in the OP, Noson S. Yanofsky, from 2003 called A Universal Approach to Self-Referential Paradoxes, Incompleteness and Fixed Points. Yanofsky has tried to make the paper to be as easy to read as possible and admits that when abstaining from category theory, there might be something missing. However it's a very interesting paper.

In it he makes very interesting remarks:

On a philosophical level, this generalized Cantor’s theorem says that as long
as the truth-values or properties of T are non-trivial, there is no way that a
set T of things can “talk about” or “describe” their own truthfulness or their
own properties. In other words, there must be a limitation in the way that T
deals with its own properties. The Liar paradox is the three thousand year-old
primary example that shows that natural languages should not talk about their
own truthfulness. Russell’s paradox shows that naive set theory is inherently
flawed because sets can talk about their own properties (membership.) Gödel’s
incompleteness results shows that arithmetic can not talk completely about
its own provability. Turing’s Halting problem shows that computers can not
completely deal with the property of whether a computer will halt or go into
an infinite loop. All these different examples are really saying the same thing:
there will be trouble when things deal with their own properties. It is with this
in mind that we try to make a single formalism that describes all these diverse
– yet similar – ideas.

The best part of this unified scheme is that it shows that there are really no
paradoxes. There are limitations. Paradoxes are ways of showing that if you
permit one to violate a limitation, then you will get an inconsistent systems.

And I would really underline the last chapter above. The issue is about limitations and if you end up in a paradox, you simply have had an inconsistent system to start with. Usually in the way that your premises or the "axioms" you have held to be obviously true, aren't actually true, not at least in every case. Hence an outcome similar to Russell's paradox is simply a logical consequence of this. Also understanding that these are limitations doesn't mean that the consistency of mathematics is brought to question. I think on the contrary: you simply have to have these kind of limitations for mathematics to be logical and consistent.

(If anybody is interested, there are some classes by Yanofsky in Youtube, for example Outer limits of reason. I haven't watched them yet, so I cannot rate them.)

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.

IMO those concepts are far too subtle to be introduced the first day of foundations class. Depending on the level of the class, I suppose. Let alone "Introduction to mathematics," which sounds like a class for liberal arts students to satisfy a science requirement without subjecting them to the traditional math or engineering curricula. — fishfry

There's a lot that in mathematics is simply mentioned, perhaps a proof is given, and then the course moves forward. And yes, perhaps the more better course would be the "philosophy of mathematics" or the "introduction to the philosophy of mathematics". So I think this forum is actually a perfect spot for discussion about this.

Of course it would be a natural start when starting to talk about mathematics, just as when I was on the First Grade in Finland the educational system then had this wonderful idea of starting to teach first grade math starting with ...set theory and sets. Ok, I then understood the pictures of sets, but imagine first graders trying to grasp injections, surjections and bijections as the first thing to learn about math. I remember showing my first math book to my grand father who was a math teacher and his response was "Oh, that's way too hard for children like you." Few years later they dropped this courageous attempt to modernize math teaching for kids and went backt to the "old school" way of starting with addition of small natural numbers with perhaps some drawings and references about a numbers being sets. (Yeah, simply learning by heart to add, subtract, multiply and divide by the natural numbers up to 10 is something that actually everybody needs to know.)

Truth versus provability is not a suitable topic near the beginning of anyone's math journey. IMO of course. — fishfry

It sure is interesting. And fitting to a forum like this. If you know good books that ponder the similarity or difference of the two, please tell.

I'm not sure how the subject came up. — fishfry

From the OP at least I made the connection.

It's interesting to know that all these diagonal type proofs can be abstracted to a common structure. They are all saying the same thing. — fishfry

That's what really intrigues me. Especially when you look at how famous and still puzzling these proofs are...or the paradoxes. Just look at what is given as corollaries to Lawvere's fixed point theorem:

Cantor's theorem
Cantor's diagonal argument
Diagonal lemma
Russell's paradox
Gödel's first incompleteness theorem
Tarski's undefinability theorem
Turing's proof
Löb's paradox
Roger's fixed-point theorem
Rice's theorem

Of course in mathematics a lot theorems have corollaries, but I would just point out to what these theorems are about: limitations in proving, limitations in computation and a paradox, that basically ruined naive set theory and spurred the creation of ZF-logic. All coming from a rather simple thing.

Going back to the OP and the article given there, perhaps in the future it will be totally natural (or perhaps it is already) to start a foundation of mathematics or a introduction to mathematics -course with a Venn diagram that Yanofsky has page 4 has. Then give that 5 to 15 minutes of philosophical attention to it and then move to obvious section of mathematics, the computable and provable part.

Thank you, @fishfry

It seems that from you I get extremely good answers. Yes, Lawvere's fixed point theorem was exactly the kind of result that I was looking for. It's just typical that when the collories are discussed themselves, no mention of this. I'll then have to read what Lawvere has written about this.

And that not necessary is important for me. This is what @TonesInDeepFreeze was pointing out to me also. I'll correct my wording on this.

Thanks for the educative response. But it seems that you went back to my earlier post.

So just to make things clear, I'll ask again:

But if you start from that there is no bijection, and then prove it by:
If there is a bijection then there is a surjection
There is no surjection.
Therefore, there is no bijection.

Isn't that a proof by contradiction? — ssu

Now why I'm ranting so much about negative self-reference or diagonalization, which I acknowledge I haven't accurately defined, is that it crops so easily in many important findings. Yet what is lacking is a general definition.

Here's a video explaining this perhaps better than me:

Could this be put to even more simple terms?

Wrong.

Aren't you forgetting the oldest monotheistic religion, the one of the oldest Empires and Rome's old nemesis, the Fire worshipping Persians? Zoroastrianism is the oldest monotheistic religion as it is roughly 500 years older than the Jewish religion. But because Islam conquered the Sassanid Empire, we don't hear much about them. But there are a few still even today alive and worshipping the old religion.



I'm always fascinated by the idea that even if there wouldn't be Islam, or the Sassanid and Byzantine empires had stopped the spread of Islam, Iranian still today would be seen as different from us, the Western people as likely they would be all Zoroastrians.

It can be interesting to consider how far philosophy/rationality can lead us towards an understanding of God. Perhaps some type of prime mover necessarily exists. — BitconnectCarlos

The study of religion is bit different from the attempt to prove God's existence. The questioning doesn't even start from the obvious question: Is there a God?

Why would continental Europe agree to that? — Tzeentch

If the US walks away from Europe, then naturally Continental Europe would love to have the support from the UK. Two aircraft carriers are always welcome.

Does infinity actually mean that there is always one more, or does it just mean the possibility of it? — Sir2u

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.)

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. — Metaphysician Undercover

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.

I agree roughly with what you wrote, but aren't you going a little light on the UK?

It was their errand boy that went to Ukraine to boycot peace, acting diametrically against Ukrainian and European interests to score brownie points with the Americans. — Tzeentch

During Trump's office, the British Parliament understood quite clearly that if Trump really walks out of NATO, they have to take more role in Continental Europa. I don't think that has changed, from the tanks that the UK had, Challenger tanks are now in Ukraine. Sure, UK wants to be the closest ally of the US, but Trump will shit on every ally it has, except Isreal. In the case of Israel, the US is it's ally, not the other way around. This isn't because of the Jewish Americans voters, but because of the many millions more of pro-Israeli Christian voters in the US.

Especially from a Finn I would expect a certain critical stance towards those pushing for war, since your nation will be on the frontline paying the heaviest price if the worst comes to pass. — Tzeentch

My friend @Tzeentch, we have discussed much in the Ukraine, and if this thread comes too popular or the heated, likely it will whisked away to the Lounge as the Ukraine conflict -thread.

But to others, the actual story both Sweden and Finland did everything to keep the relations normal with the cranky neighbor in the East. And we really did, the whole term of Finlandization was invented for the Finnish situation. But there's a point until you try to be neutral and cordial and keeping up friendly relations to your cranky threatening neighbor. That point was crossed over in February 24th 2022. That it was it. Finland and Sweden abandoned both their neutrality, as Russia is obviously a threat to them.

I'm writing this just a few kilometers from the Russian border. There is NO traffic over the border, for years I haven't seen a single Russian truck and if you want to go to Russia, you have to go through Turkey. The Finnish Armed Forces have put the training cycle to a totally different gear to prop up the deterrence. Russia is spreading bullshit propaganda to it's people over the border that Finland is planning to invade Russian Karelia. Russian Foreign minister Sergei Lavrov yearns for the days of Finlandization and talks about it's opposite "Estonization" which for the Lavrov means Russophobia. (See here)

Well, you reap what you sow.

In my view, Europeans should not focus on which clown is driving the clown car, nor on anything the clowns are saying.

The only thing that matters is Washington's actions, and what we can reasonably glean to be Washington's interests in order to predict their future actions. — Tzeentch

I agree with this, with the addition that perhaps we should listen what the US is saying and try do cooperate with country. The boisterous rhetoric of Trump can be put into one category, it's basically intended for his own base, the actual actions are another issue.

Godel wrote his proof of God for the same reason as why he wrote all his other proofs: because he could. — Tarskian

Notice that he wasn't an atheist and he did believe in God.

Because it's stupid and pointless if there is no God. — bert1

Really?
Acting righteously, being good to other people all those things...you do because only because of God?
And without God it would be pointless?

This has become more actual again now that Biden turns out to be a demented nutjob holding onto power for no apparent good reason, making sure the Democrats will lose. Now that Trump is pretty much a shoe in, what should the EU do and what can we expect with respect to, for instance, Ukraine?

@ssu @Tzeentch thoughts? — Benkei

First of all, Biden has been already for a long time a demented politician holding for power and totally incapable of seeing that he himself is not up to the job. Let's not kid ourselves with that.

Trump has been leading the polls and with this pace, he will likely be the next president. But still much can happen.

In my view, the best thing is to just SHUT UP and calmly observe the garbage fire called the 2024 elections. Respect the US political process enough to not to intervene, at least not publicly. The worst thing would be for the EU to publicly support Biden. That would anger many of those that won't vote for Trump and simply offend Americans and likely it will offend Trump even more. Then understand that you will have a weak US administration entangled with domestic problem whatever happens. Only react if Trump attempts to do changes to the relationship, which he surely will do. Then don't budge. Trump is a bully and his followers just love how "the establishment" hates him. Yet he really is no Hitler. His real chance for a self-coup went as he couldn't even control his own security staff that drove him to the White House and not to Capitol Hill as he wanted during that infamous day.

If Trump wins, you will have an more erratic administration as earlier as Trump will likely bring on far more of his totally loyal yes-men into his administration. This actually didn't happen in the first administration: Trump for example brought on totally reasonable military officers (with the exception of Mike Flynn, who lasted 24 days) and they made Trump's foreign policy to be quite standard US policy (what else?). Trump is a great populist, but he lacks leadership qualities. He has the attention span of a five year old, if the topic isn't about himself. Yet after being left alone in the Florida mansion, I think Trump has a lot of built up anger and he has been contemplating "the next time..." for years now. And even if he know does have the best A-team that Republicans can offer, he can be and will be Trump as he is unmanageable. And Trump can promote his most loyal nutjob followers to prominent positions to make a total shit show.

With Biden winning, if that would happen, then it's old-Brezhnev-time for the US where you cannot know who actually is in charge. But it's a collective effort. And if Biden would come down or is hospitalized, then the Democrats have the opportunity to get a candidate that could win Trump.

The only thing here for the EU is to check really it's defense policy and in this field make more cooperation with the UK. As the UK never did leave NATO, defense cooperation would be a natural start for the EU to warm ties with the UK. Yes, NATO isn't in any way attached to the EU, but here the lines organization lines don't matter so much and Austria wouldn't mind.

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: — javi2541997

Not under the assumption that quantities are unlimited. — Metaphysician Undercover

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? — Sir2u

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.

Act righteously and divine favor will follow. "Reasoning God's existence" is not a biblical concern at all. — BitconnectCarlos

Exactly. So I'm puzzled by those who want to give a proof of God, because they usually are religious people. Why not simply follow the given manuals and act righteously?

It's garden variety modus tollens:

If there is a bijection then there is a surjection
There is no surjection.
Therefore, there is no bijection.

No need for a reductio assumption. — TonesInDeepFreeze

Yes,

But if you start from that there is no bijection, and then prove it by:
If there is a bijection then there is a surjection
There is no surjection.
Therefore, there is no bijection.

Isn't that a proof by contradiction. That was my point.

And you see now that a reductio argument is not needed; indeed Cantor did not use a reductio argument. — TonesInDeepFreeze

OK, so let me try get your viewpoint here: having the list g and constructing the real that is not on the list isn't itself using reductio ad absurdum. Yes, this obvious to me also.

However, I still insist that to prove that there's no 1-to-1 correspondence between exist between the natural numbers and the reals is a proof by contradiction, where you use what was done above. So when I talked about diagonalization, being an amateur here, I also referred to consequences and this (the list, anti-diagonal construct which isn't on the list) being used as part of a wider proof. For you diagonalization is just the part with the list g and the construction of the real that isn't in it. I can understand that totally.

Either this is the issue, or then I have to try to spend even more of your precious time.

:grin:

He's assumed to exist. — BitconnectCarlos

And what is his follower assumed to do? To reason God's existence? Or perhaps to do something else?

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. — Metaphysician Undercover

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.

Ok, I think I now understood you. By g you referred to the list, while I thought you meant the constructed real that isn't on g.

I don't say that it is wrong. I just say that it is highly confusing. — Tarskian

Math is confusing. It's far more closer to philosophy than mathematicians and logicians want to admit.

For example, I've followed Chaitin's story and his efforts with the Omegan number and AIT, on and off for now about 20 years and seen how he has been gotten even ad hominem attacks (actually from my own university, from where I graduated). Only now it seems that Gregory Chaitin is getting respect.

OP is another crank (like PL) hiding behind fancy mathematical and logical language to push his nonsense, this time the nonsense being religious proselytising, as can be seen from his other posts. — Lionino

I wouldn't go for ad hominems, but for me this thread is informative. So hopefully nobody is banned and the tempers don't rise too much.

I myself follow the rule that if two or more PF members saying you are wrong (not just that they oppose you view in some way) with nobody agreeing with you, then you might really look sincerely where this error would lies.

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. — Metaphysician Undercover

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. — Metaphysician Undercover

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. — Metaphysician Undercover

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.

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. — Metaphysician Undercover

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).

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. — javi2541997

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? — javi2541997

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.

No, as I said, Cantor did not make that reductio assumption. Again:

Let g be an arbitrary list of denumerable binary sequences. (We do NOT need to ASSUME that this is a list of ALL the denumerable binary sequences). Then we show that g is not a list of all the denumerable binary sequences. — TonesInDeepFreeze

Well, in my example (which is common), I was referring to reals between 0 an 1, not ALL reals.

I think that we aren't understanding each other here:

If you say " Then we show that g is not a list of all the denumerable binary sequences." Isn't then that g is not in the list of all these sequences exactly constructed by diagonalization? And here the negative self-reference is that g is not on this list, the list of (in your version) all the denumerable binary sequences.

Turing constructed a quite important and remarkable proof for the uncomputability of the Entscheidungsproblem. But is that constructiveness a problem? — ssu

Church is the one who addressed the Entscheidungsproblem. Turing proved the unsolvability of the halting problem. — TonesInDeepFreeze

Not quite, even if it's great that someone remembers Church's role (although he is remembered by us referring to Church-Turing thesis). Alan Turing's paper is called "ON COMPUTABLE NUMBERS, WITH AN APPLICATION TO THE ENTSCHEIDUNGSPROBLEM".

Where Turing states:

Although the class of computable numbers is so great, and in many
ways similar to the class of real numbers, it is nevertheless enumerable.
In §8 I examine certain arguments which would seem to prove the contrary.
By the correct application of one of these arguments, conclusions are
reached which are superficially similar to those of Gödel. These results
have valuable applications. In particular, it is shown (§11) that the
Hilbertian Entscheidungsproblem can have no solution.

It's Turings paper itself where we get "the halting problem" of a Turing Machine (in the paper referred to a machine. The name "Turing Machine" was given by his teacher, Alonzo Church).

Thanks for answering my ramblings, @TonesInDeepFreeze,

A sentence is provable from a set of axioms and set of inference rules if and only if there is a proof sequence (or tree, tableaux, etc.) resulting with the statement. — TonesInDeepFreeze

As an non-mathematician/logician, I'm not familiar with the terminology. So it is sentence - proof sequence - axioms? I still assume there is a link between sentence and the set of axioms.

No, diagonalization does not require indirect proof. — TonesInDeepFreeze

Diagonalization itself of course doesn't require an indirect proof. What I meant that it itself is an indirect proof: first is assumed that all reals, lets say on the range, (0 to 1) can be listed and from this list through diagonalization is a made a real that is cannot be on the list. Hence not all the reals can be listed and hence no 1-to-1 correspondence with natural numbers. Reductio ad absurdum.

Perhaps using the "negative self-reference" would be better if the reductio ad absurdum proof isn't exactly about changing something on a diagonal.

What the Turing Machine cannot compute is found exactly by using diagonalization (or negative self-reference) that we are talking in the first place. — ssu

I think the theorem you have in mind is that there is no algorithm that decides whether a program and input halt. The proof uses diagonalization. — TonesInDeepFreeze

Exactly.

But, again, the proof is constructive. Given an algorithm, we construct a program and input such that the algorithm does not decide whether the program halts with that input. — TonesInDeepFreeze

Yes. Obviously Turing constructed a quite important and remarkable proof for the uncomputability of the Entscheidungsproblem. But is that constructiveness a problem?

What are you trying to get at? — Metaphysician Undercover

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.

Is the 100% of the food is for 100% of the dogs. — Sir2u

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

It makes no difference the actual quantity of the food, only the correspondence of food to dogs. — Sir2u

I think so. As I said: if you double the amount of food to every dog, it doesn't matter as they can be only measured to each other. There would be no difference. Notice that measuring is possible with the random dog that Plato picked up. Yet If you give all the dogs just the amount as Plato's picked up dog eats, that would leave a lot of dogs hungry and a lot with way more food they eat. That would create a mess.

Doesn't mathematics start with the unit, one, as the point of comparison, just like Plato\s dog — Metaphysician Undercover

I'm not so sure that mathematics starts from exactly one thing. :smile:

The actual problem is when we try to measure the system of measurement. — Metaphysician Undercover

Well, think in the story about how much all dogs eat, then remember the rules.

The starting point, "Plato's dog" is a limitation on the act of measuring, imposed by choice, it is not a limitation on any dogs. — Metaphysician Undercover

On the other hand, with Plato's dog, we can do something as important as count and measure. The first thing that mathematics evolved from, and something that smart animals can also in their way do.

And in my view this measurement creates the confusion. Here with dogs that simply cannot be measured as their definition relies on this (if you could measure it, they wouldn't eat the least or most as Plato is totally right in this way). And I think this is the problem when we want to view mathematics as a logical system, but start from the natural numbers and assume something like addition is a meaningful operation with everything. Yet Mathematics, as a logical system, holds true mathematical objects that aren't countable or directly provable. I think we are still missing something very essential here.

Any readable proof of Cantor's Theorem will contain at most a finite number of characters. — Banno

Any proof will contain at most a finite number of characters. At least for us finite entities.

Yet it shows that there are numbers with a cardinality greater than ℵ0. — Banno

That's actually not Cantor's theorem (the power set of any set has a strictly greater cardinality than the set itself).

What Cantor shows is that there cannot be a bijection between the natural numbers and the reals by reductio ad absurdum. That's it. Notice that it's an indirect proof. And notice that already from this we have an open question, the Continuum Hypothesis.

Yet this doesn't stop Cantor treating uncountable infinities as normal ones and he continues adding things up in cascading system of larger and larger infinities while trying to evade paradoxes. Many mathematicians even today are doubtful of this, even if they might be not mainstream.

I don't understand what you're saying here. Can you explain? — Metaphysician Undercover

Ok, If you start from Plato's dog as the measure for all dogs, let's call it dog 1, you get dog 2 (that eats twice the amount), dog 3 (eating triple amount), dog 4 and so on. And obviously for any dog n, then there's a dog n+1 and so on. And from this, in reality Aristotle (not Plato, in reality) would talk about only a potential infinity. And this idea stayed until Cantor, for example Gauss wrote in 1831: “I protest against the use of infinite magnitude as something completed, which is never permissible in mathematics. Infinity is merely a way of speaking” and Kronecker, who vigorously disagreed at Cantor famously said "God created the integers, all the rest is the work of Man".

This is what I tried to refer to, when I said " it's a limitation, when you start from Plato's dog." Perhaps better wording would be simply a rejection.

I think others have here explained why in the story dog eating the most and least aren't limitations starting from @Sir2u, @L'éléphant and @Lucky R.

But don't worry, this is only perhaps the most heated and difficult issue in mathematics, ever.

It is doubtful that the food could be broken down to anything less that a molecule and still be counted as food even though the food is dividable. That would be the food for the dog that ate the least. And because the food is dividable to share amongst the little beasts, that would limit the amount that could be eaten by the one that ate the most. — Sir2u

What I was trying to say with rule 3, things like physical dimensions or other physical aspects wouldn't be taken into question (as Viking dogs do also take space and also Ancient Greece had a limited area) as the amount of food the dogs eat can be compared to only to the dogs.

No dog could eat all of the food as there would be none for the rest of them. — Sir2u

Hope you here notice the incommensurability between what the dog that eats more than everybody else and any dog that can be measured by Plato's picked up dog. And what is "all the food" for the dogs since the food can be compared among the dogs? You cannot double the food amount of all dogs, or take half the food away from every dog. Just as with whatever dog Plato picks up, he'll by his definition pick up Dog 1. There's enough of food, the goddesses made sure about that.

Yes. That will work fine if the criterion for their order can't change. But you have posited that they can change how much they eat. — Ludwig V

Hopefully I didn't. All the dogs eat exactly a defined amount of food different from any other dog, not less, not more.

(BTW, if someone is puzzled why I did choose Plato and not Aristotle, it's because we learn about Zeno of Elea through Plato's writings. Of course the objection is more Aristotle, but the time gap is even bigger between Zeno and Aristotle. Some reality to a story with Greek goddesses meeting philosophers. :joke: )

Zeno is right. Not by reason of counting. 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.

It is always valid to say "there is at least one dog that eats the most" and "there is at least one dog that eats the least". — L'éléphant

I agree. I had identical thoughts, but I couldn't find the perfect words to express them as you did. :sweat:
Yes, I am one of the 60% of voters that chose the second choice. — javi2541997

Absolutely fantastic! :grin:

And yes, #2 gives us the opportunity to do this. Not by reason of counting as obviously both of Zeno's dogs are literally uncountable. Notice how different the story would come if Athena would have said: "By using this dog and what it eats, put all the other dogs into order."

As with the story, I argue that this has been a real problem that Math at least over 2400 years.

Now the thing is that our confusion hasn't stopped us for 2400 years as we are already using both of Zeno's dogs in a variety of way in mathematics, but since this question isn't answered how Plato's and Zeno's dog should logically coexist, we simply have different names. In the case of the dog that eats the least, usually called an infinitesimal, we have non-standard number systems (surreal and hyperreal numbers) and Non-standard analysis.

And not only that. When you think that in the story "the least eating dog's meal" < "every other dogs meal", think about how Dedekind cuts are used to define real numbers. The name's cut comes from using greater than and less than signs ">" and "<".

Yet the other Zeno's dog and it's problematic diet comes immediately into question when people tried to form the foundations of mathematics like with set theory. The great Cantor understood what in the story is Plato's argument. For example, when he had the idea of sets having Power set, which are bigger than the set itself, he understood how the power set for "set of all sets" is a bit problematic. And thus Cantor's set theory is hierarchial. Yet what is interesting (and what I can thank the PF forum leading me to a great book) is that Cantor didn't reject the notion of Absolute Infinity, or what in the story would be the dog that eats the most. As he couldn't describe this infinity, he thought it was something that God would know. Or God. Something that for a deeply religious person is something important (unlike for the many reading Cantor's texts). Still, with Frege's simple idea of Basic Law V invites immediately the other of Zeno's dogs to the picture.

As you can notice, in the poll of the story I didn't actually ask if Plato is right or both are right. And this is the real question for mathematics: how can both Zeno's dogs and Plato's dogs coexist in peace?