Ok, so what's the interesting thing with having both addition and multiplication?

Yanofsky points out that only a very small part of Th(N), i.e. arithmetical truth, is provable. The remainder of Th(N) is unpredictable and chaotic. Most of Th(N) is even ineffable. — Tarskian

With a formal system with Peano Arithmetic we already get the results of Gödel's incompleteness. Hence this has been shown earlier than Yanofsky's paper. Yet do notice that Presburger Arithmetic is complete.

So what's the thing with multiplication?

Well, a shitty peace deal is all the Ukrainians will be getting and they have the US and cronies to thank for it. — Tzeentch

Without any help from the West Russia would have likely obtained it's objectives. Which would have been even more shitty for the country. Likely they would have lost the coast to the Black Sea.

The Kremlin has signaled that they want a diplomatic settlement since the start of the war. — Tzeentch

With denazification and all that? Lol.

Once Trump enters office that will be off the table, and he will likely be free to force Ukraine to sign an uncomfortable peace deal with the Russians or withdraw support. — Tzeentch

That's what I was writing about. Trump makes absolutely shitty peace deals. The peace deal with the Taleban was really surrender, which then Biden gladly accepted (and hence there's absolutely no discussion of this defeat as both parties are culprits to the lost war). I bet that Kim Il Sung would have gladly accepted a similar peace terms, and if South Korea would have been left to face North Korea and China alone, I'm sure that there would be unified Korea, just as there's a Vietnam today.

After that, the Russians will in all likelihood seek a return to the pre-2014 status quo, restoring economic ties with Europe. — Tzeentch

Good luck with that. Only when Putin is dead and buried perhaps something like that can happen.

They have no reason to involve themselves into large-scale conflict with Europe when the US and China are on the cusp of war, and with Europe and Russia standing to profit greatly from that conflict. — Tzeentch

Russia wants Finlandization of all Europe. And if the US "pivot people" get their way and US really "pivots" to Asia (what that means I don't know as the US is already in Asia) and doesn't care Europe anymore and the EU doesn't hold together, then Russia can pick every European country one-by-one. Russia is far more powerful than any European nation on it's own. Hence it's no surprise that Russia wants to break the Atlantic tie.

Europe doesn't profit from a US China war. Russia does. Anything that's worse for the US is good for Putin's Russia.

The "why" here leads right to physics, and the natural sciences more broadly, because a big part of the "why" seems to involve how our symbolic systems have an extremely useful correspondence to how the "physical world" is. — Count Timothy von Icarus

Aren't these symbolic systems of mathematics extremely useful in the US elections too? Isn't counting the votes quite essential in free and fair elections?

But the question "why do we do this?" leads right to questions about "how the world is" which tend to include physics and metaphysics. — Count Timothy von Icarus

And that's why reporters ask metaphysical questions from cosmologists or quantum-physicists and not from philosophers, who actually could be far more knowledgeable about metaphysical questions.

Yes, I totally understand the arguments of mathematics being an essential tool for physics and physics is an inspiration to create new mathematics and this all leads to reductionism of physics and math.

However, why do we stop there? Or to put it in another way, why then the rejection of what is quite important to us, the society and the World humans have built for themselves and which is studied by the humanities/social sciences in academia?

Let's remember topic of the thread and the idea that there's non-computable mathematics: that many true mathematical statements aren't provable or computable. How do we get to those things that are not computable, not provable? As discussed here in the OP and then later in the discussion of Lawvere's theorem in Category theory, many of these theorems showing the limitations of mathematics have self-reference and diagonalization in their argument. Negative self-refence seems to be a limit for computation.

Now, just ask yourself: We base a lot of our actions on past history. And we also try to learn from our past mistakes even as a collective, that we don't the same mistakes as in the past. Wouldn't that be perfectly modeled by negative self-reference? If so, then could you then argue that historians don't explain history by computing functions because their field of study falls into non-computable mathematics? Without computabilty, the only thing might be left is a narrative explanation of what happened.

And please understand, my argument is that indeed everything is mathematical, when we want to be logical.

I'm always intrigued why a conversation about math morphs to conversation about physics.

Why wouldn't a discussion of mathematics morph into a conversation about the US elections? In elections mathematics plays a pivotal part too: who gets the largest number of votes. And when you have these different kinds of electoral systems, then it can happen that the candidate that gets the most vost votes isn't actually elected. Yet elections are math, aren't they? :wink:

Trump has vowed that he could end the war in Ukraine in "one day" when President. Even if it's the ordinary populist Trumpian rhetoric from Trump, we have to look at what his last surrender peace deal was with the Taliban: all that Taliban needed to do was not to attack Americans, yet they could basically were given a free hand to attack the pro-American government as they wanted. Quite a similar to the Dolchstoss that Americans gave to the South Vietnamese in the 1970's. But if you want a deal at all costs, that's the kind of shitty peace deal you will get along with the fact that you lost your longest war you had fought.

End result, Trump will make his grandiose attempt for a peace deal, which very likely it will fail. Trump angrily will want to cut all support to Ukraine, but Pentagon and the Republic Party won't accept this, and Trump will end up cutting the aid to Ukraine. For the Russians this war isn't a sideshow from where to "pivot" somewhere else as for the Americans, hence if they aren't fought to a standstill, they'll continue the war. Russia will halt the war only if continuing the war leads to a far more worse outcome. This should be understood from the Russians.

Europe should understand that for at least 4 years with Trump the US will be a very unpredictable ally and they have to put money on defense and support Ukraine themselves (hopefully increase the aid).

First it was Schiff, then it was Chuck Schumer who have pushed to Biden to give up his candidacy.
But the Democrats have really a problem even this way, because in this age of DEI, they simply cannot bypass a black female vice president.

Well, the Democrats will loose just like the Conservative party lost in the UK. Perhaps it's not going to be such a loss as the Conservatives had (worst election defeat in 190 years), but still.

My intuition says that Yanofsky is probably right, and that it is Cantor's theorem that is at the root of it all, but I am currently still struggling with the details of what he writes. — Tarskian

I agree, also with Yanofsky.

Cantor's proof is the simplest form of diagonalization that has all the "problematic" consequences, once we start to look at infinite sets (with finite sets Cantor's theorem is quite trivial). As Yanofsky say's:

If you try to express all the truth about the natural numbers, you are effectively trying to create an onto mapping between the natural numbers and its power set, the real numbers, in violation of Cantor's theorem.

And of course, with the proof of the theorem, using diagonalization, we showed that a surjection / onto mapping is not possible. This shows just how close making a bijection is to giving a proof. We understand that an infinite set is incommensurable to a finite set and that we cannot count finite numbers and get to infinity. However, this isn't the only thing we have problems once we encounter the infinite.

After all, if a formal system can express Peano Arithmetic, then Gödel's second incompleteness theorem holds that the system cannot prove it's own consistency.

Perhaps Berkeley had a point. Perhaps the concept of incommensurability could help here? — Ludwig V

Definitely.

It should be obvious that with infinity or anything infinite, you have incommensurability that you don't have when just handling finite numbers. But once you have incommensurability, what else you don't have?

Thanks to both of you. And no, it isn't nitpicking. Of course we can talk about surjective or injective functions. What for me it's very irritating that there aren't these general definitions. As a layman I would think that something being an equation, a mathematical statement that shows two or more amounts are equal, would also be a (or could be modeled as a) bijection. But, uh, apparently not. :(

And we haven't even discussed isomorphisms and their relation to bijections. Perhaps it's better simply to talk about bijections, injections and surjections. At least that ought to be simple, I hope. Far more easier than these than to talk about Turing Machines, or (yikes), Gödel numbers!

And if that was the only thing correcting, then I'm not totally wrong in the discussion. :)

If Biden continues to be Democratic candidate, I think after yesterday Trump will very likely will be the next President. Now way to deny this photo by Evan Vucci will be a historical one:



Everybody, just think how fucking long this thread will be! :yikes:

Now on page 734 and then still going on until 2028 or so...

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.

This comment is typical. It is very sharp, very pointed. But the calculus is embedded in our science and technology. — Ludwig V

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.

Of course engineers don't care shit about logical foundations if something simply works and is a great tool.

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

To my reasoning it doesn't. And both Leibniz and Newton could simply discard them too with similar logic.

What do you mean by "actual infinity"? — Ludwig V

I'll give the definition from earlier:

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.

For example Cantor uses actual infinity as the talks about the set of natural numbers being the same size that rational numbers, yet them being smaller than the real numbers. All of these sets are of finished "actual infinity", not the potential infinity as the Greeks thought.

While Cantor says something simple, i.e. any onto mapping of a set onto its power set will fail, Yanofsky says something much more general that I do not fully grasp. — Tarskian

Ok, this is very important and seemingly easy, but a really difficult issue altogether. So I'll give my 5 cents, but if anyone finds a mistake, please correct me.

Let's first think about how truly important in mathematics is making a bijection, which is both an injection and a surjection. We can call it a 1 to 1 correspondence or a 1-to-1 mapping. And basically bijections are equations like y=f(x) or 1+1=2. And of course Cantor found the way to measure infinite sets by making bijections between them, like there's a bijection between the natural numbers N and the rational numbers Q.

With the diagonal argument or diagonalization, by negative self-reference we show that a bijection is impossible to make as the relation is not surjective. This is the proof for Cantor's theorem. Yet this is also the general issue that Yanofsky is talking about as this is found on all of these theorems.

Even in the case of your example in the OP (if I have understand correctly, that is) first it is assumed that the Oracle can make a bijection from the past to the future and hence can make correct predictions about everything. Then with the Thwarter app, because of the negative self-reference, means that the situation for the Oracle is that it cannot make a bijection as the new situation with the Thwarter app is not surjective anymore.

And as @noAxioms immediately pointed out, you are basically using Turing's proof in your model. Which itself uses also diagonalization.

Hopefully this was useful for you.

I'm afraid I don't know what "^" means. — Ludwig V

Writing x^2 means x². A bit lazy to use this way of writing the equation.

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

Exactly. With limits we want to avoid this trouble. Yet it isn't actually a paradox as infinitesimals are rigorous in non-standard analysis.

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

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):

“They are neither finite Quantities nor Quantities infinitely small, nor yet nothing. May we not call them the Ghosts of departed Quantities?

Right from the beginning, 2,500 years ago, people have been thinking that everything has been done and is perfect. — Ludwig V

I agree. Perhaps they admit that there's just only some minor details missing, that aren't so important.

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

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.

Why do I say so?

Let's take the case how Set theory gives us the actual infinity and various sizes of infinity with Cantor's theorem. What in Cantor's theorem is used is Cantor's diagonalization (or Cantor's diagonal argument). 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. We get things like:

Russell's paradox
Gödel's first incompleteness theorem
Tarski's undefinability theorem
Turing's proof
Löb's paradox

These all actually tell us of limitations. And hence it shouldn't be any wonder that if we talk about Zeno's dogs, there are obvious limitations to our finite reasoning.

If a program knows a list of things it can do [ A1, A2, A3, ..., An], and it receives the instruction "do something else but not Ak", then it can randomly pick any action from [A1, A2, ..., A(k-1),A(k+1) .... An] as long as it is not Ak. — Tarskian

Randomly picking some action from [A1, A2, ..., A(k-1),A(k+1) .... An] as long as it is not Ak is surely not "do something else". It is an exact order that is in the program that the Oracle can surely know. Just like "If Ak" then take "Ak+1". A computer or Turing Machine cannot do something not described in it's program.

Believe it or not, I can see that. — Ludwig V

:grin:

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

Both Newton and Leibniz figured out the way to make a derivation by using infinitesimals.

Let's say that we want to make a derivation of x^2 = 2x With infinitesimals it goes like this:
If dx is an infinitesimal change in x, then the corresponding change in y is dy = (x+dx)^2 - x^2, so

dy/dx = (x+dx)^2 - x^2 / dx = 2x(dx)+(dx)^2 / dx = 2x + dx

And because dx is so infinitesimally small, then we can ignore it and dy/dx = 2x.

And here's the problem: if we just ignore dx, then it would be zero, right?. But then again, we cannot divide by zero! So it has to be larger than zero, but then it also has to act as zero. That's the confusion and And this is actually similar what problem I stated earlier: 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! (Remember, 1=0,9999...) Because if it would have decreased, then obviously this amount could be divided into smaller amounts.

And hence we use limits.

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

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.

Frege proposed a way that it would be a logical truth. But his way was inconsistent. — TonesInDeepFreeze

Isn't that a bit too much to put on the Basic Law V?
If we have problems with infinite sets, why would you throw away also everything finite?

How about Peano axioms or Peano Arithmetic?
Are they inconsistent also according to you?

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

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:

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.

This meant that there was no limitations on what a set could have inside it and Russel could then form "the set of all sets that do not contain themselves as elements", which is a contradiction. Yet notice the problems of Zeno's dogs had already been found when thinking of the set of all sets. There was the Burali-Forti paradox of the largest ordinal (explained earlier) and what is named Cantor's paradox of there not existing a set of all cardinalities (hence Cantor understood that if set of all cardinalities is accepted, then what would be the cardinality of this set?). This simply goes back to in the story of Plato's rejection of Zeno's most eating dog, just in a different form.

And basically what is lacking here is that with Zeno's dogs addition simply doesn't have an effect. This is why idea of infinitesimals is rejected in standard analysis. Because these infinitesimals cannot be used as normal numbers.

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. And that's the topic of

1 = 0,999999...

Ok, if modeled into the story, you could then find the least eating Zeno's dog eating it's meager rations in the end of that line depicted with "...". OK, why has this be exactly equal to one? Well, if we would assume that

1 > 0,999999...

This would simple mean that Zeno's infinitesimal dog would eat a finite amount, and hence it wouldn't be the least eating dog as Plato's arguing is true about the finite is never ending. With the infinite, ordinary arithmetic breaks down.

So basically the problem is that Zeno's dogs, what I could dare to call infinitesimal and Absolute Infinity, are obscure mathematical entities (and even quite heretical entities) as we don't have the idea just how normal arithmetic breaks down and how then they could be part of "the other dogs". Hence I would state that there's something missing in math.

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.

In fact, in a great presentation of how Cantorian Set Theory counts past infinity and creates larger and larger infinities is from a popular Youtuber Vsauce below. One should view it altogether as it's a good presentation, but notice just what he says about mathematics from 12:19 onward as this just shows how much mathematicians have become lawyers (or basically have outsourced the foundations of mathematics to logicians).

Thwarter needs a prediction as input. Otherwise it does not run. — Tarskian

Yes, But notice that the Oracle staying silent can be also viewed as an input. So when the Oracle is silent and doesn't make a prediction, the Thwarter can do something (perhaps mock the Oracle's limited abilities to make predictions), which should be easily predictable.

Yes, of course, Oracle can perfectly know what is truly going to happen. However, his knowledge of the truth is not actionable. — Tarskian

Oracle can know perfectly what is going to happen if your Thwarter app is a Turing Machine that runs on a program that tells exactly how Thwarter will act on the Oracle's prediction.

And this is why you have to go a step forward from just declaring what that the Thwarter has free will. After all, what's the "free will" in the following?

Oracle predicts A -> Thwarter does B
Oracle predicts B -> Thwarter does A
Oracle predicts something else or is silent -> Thwarter does B

Notice the simple diagonalization. Now, here really both the Oracle and the Thwarter can be basically Turing Machines. Turing Machines don't have free will.

However, you do get to the really interesting point of free will when from this (which is basically a result from the Church-Turing thesis) when you make the following question: If the Oracle knows it's limitations in predicting the Thwarter, but can write Thwarter's actions down on a paper, when does the Oracle have problems even with writing the actions of the Thwarter on a paper?

The Thwarter cannot be a simple predictable program that simple reacts to the Oracle's prediction. The Oracle can easily write this down as it knows Thwarter's program.

The Thwarter app basically has to be an Oracle itself with an ability that no Turing Machine has: it has to understand it's programs it itself is running on and then change it's behaviour/action in a way that it hasn't changed ever before.

How does the Oracle now write down what is going to happen, as in this case there is not historical example of what the Thwarter will do? Well, it cannot use past information and extrapolate from it.

It should be understood here that computers cannot follow an order of "do something else". They can follow it only if in their program there's instructions what to do when asked to "do something else". And now what the "Twarter app" has to do is even more. And something doing the above, basically a "double diagonalization", if one can coin a new term.

But of course it should be evident that nobody here will crack philosophical question of free will, because the counterargument to this is that even we cannot know our own "metaprogram". Well, I would argue that as we can understand our behaviour at least partly and can learn from the past, this "double diagonalization" is at least partly something that we can do. Yet this deep philosophical question of free will won't go away.

In my view, this is an extremely important discussion, because it just shows how profound philosophical impact the findings of Turing and the Church-Turing thesis have. Just what lies beyond computability is a very important question. It's not just a limitation in mathematics for computability, it's also a deep philosophical limitation.

Comments?

You may be interested in a recent paper by Joel David Hamkins. Turing never proved the impossibility of the Halting problem! He actually proved something stronger than the Halting problem; and something else equivalent to it. But he never actually gave this commonly known proof that everyone thinks he did. Terrific, readable paper. Hamkins rocks. — fishfry

Thanks! Again a fine article, @fishfry, that I have to read. I've been listening to Youtube lectures that Joel David Hamkins gives. They are informative and understandable.

The environment of the oracle and the thwarter is perfectly deterministic. There is nothing random going on. Still, the oracle cannot ever predict correctly what is going to happen next. The oracle is therefore forced to conclude that the thwarter has free will. — Tarskian

The effects of diagonalization are important and should be discussed here in PF. It's great that this pops up in several threads and people obviously are understanding it!

Basically the oracle is similar to the Laplace's demon, that we have been talked about, for example here (real world example) in the "The Argument There Is Determinism And Free Will"-thread. One simply cannot say what one doesn't say or predict what one doesn't predict. Yet in some occasions this obviously can be the correct prediction. In your example, you make the diagonalization with the "Thwarter app".

It should be noticed that this doesn't refute determinism, it just is that any program itself or predictor himself or herself is part of the universe and once there's interaction with reality to be predicted, situations like where it cannot predict the future will happen. The pathological "Thwarter app" is similar what is describe in Turing's paper about the Entscheidungsproblem. But notice you don't have to have this app and problems will arise. (Btw, have you read Yanofsky's A Universal Approach to Self-Referential Paradoxes, Incompleteness and Fixed Points that we discussed on another thread, should be important to this too)

Yet what should be noticed is that this is a limitation that we have or any machine has in the ability to forecast everything. There's much that indeed can be accurately predicted.

And free will?

Well, this doesn't refute determinism, it's only a limitation of basically our computational abilities and logic. So the philosophical question of free will won't go anywhere.

And does the Thwarter app have free will?

Well, the thwarter app does exaclty what the original app doesn't do. Is that free will? The thwarter app still can be a program (Turing Machine) that itself cannot do something else than what is written in it's own program.

That's exactly what I have been trying to say all along! :smile: — Ludwig V

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. It really starts with the construction of von Neuman ordinals and with these you get the natural numbers. And the counting goes on in Set theory with larger and larger infinities. And when this is taken to be the building block of all mathematics, then you get into paradoxes like the Burali-Forti Paradox and to avoid the paradoxes you have to make a quite elaborate definitions like that you cannot talk about set of all sets, but of proper classes.

Now we can see just how heretical Zeno's dogs are even today for set theory, because Peano axioms give a successor function to get the next natural number and (if I'm correct) this addition to larger entities is used even with infinite quantities. Yet you cannot count to Zeno's dogs as they are basically given by an inequation: least eating dog < every other dog there exists and most eating dog > every other dog there exists. Notice that here the signs are "<" and ">" which aren't the same as "=". I'll try to explain why this is important to the story.

Let's assume A, B and C are distinct numbers and belong to the set of Natural numbers, hence they are finite. If you have the equation:

A + B = C

And if you know what two are, you will know what the third one is. So if A is two and B is three, then you know that C has to be five. But notice what happens when we change this to an inequation:

A + B < C

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.

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

Well, we can talk about the set of all natural numbers ℕ, right? I don't think that it's misleading.

Notice that it's just a model showing just how strange Zeno's dogs are. Just think of the line resembling all the dogs in a well ordered line starting from Zeno's least eat dog and ending in the dog that eats the most, you could draw it like this:

0 _____________________ ∞

Now in the line are all the finite Viking dogs. Can you pick any from the line? No, of course not. Plato's counterargument still holds. The simple fact is that if there would be a dog that eats half the amount of the dog that eats more than any other dog, then it couldn't be the dog that eats the most: we could immediately create a dog that eats more, by multiplying the "half eating dog's food" by more than 2. This is why I argue that with infinite you cannot start counting. This also shows why 1+ ∞ = ∞ and ∞ + ∞ = ∞.

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

Actually not.

The counting starts from the dog that Plato defined to be 1. The action itself defines the whole system of counting, hence the one dog that Plato picks up is always 1. Even if we assume that there really would be amounts that the dogs eat prior Plato choosing to pick up the one closest. For example, if the dog that Plato picked up would be the finite, but a large number in the octodecillion range or a bigger finite one like the one called Big Hoss, created by Jonathan Bowers, then this still wouldn't matter. You cannot increase the amount of food that the dogs eat by multiplying every dog's meal by two or by Big Hoss as the food cannot be measured anything else by the dogs.

And with Zeno's dogs you cannot count. How would you pick the next dog from the dog that eats the least? Or how would you pick a "second most" eating dog? We have to remember that Plato is correct. Just think of a finite line you draw and put at the start zero and in the end ∞ (or ω with ordinals). Between those two are all finite numbers (finite ordinals with the case of ω). Good luck trying to pick a certain finite number from the line.

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

Bravo.

In fact, what is really radical in the story is the "dog that eats the most", because current set theory doesn't accept that. Cantor said this to exist, but it was for God to know. Hence I had in the vote options the possibility "I have a different view about the whole story, ssu" in mind here.

Cantor's set theory can count the ordinals onward from ω. Yet do notice that when it then counts with infinities as like with finite numbers, it immediately (in my view at least) confronts the argument of Plato (that there cannot be an actual infinity) with the set of all ordinals and hence has get's the The Burali-Forti Paradox. Now when you think about this for a moment, that there cannot be the largest ordinal, because every ordinal has a larger ordinal number, it's quite similar to Plato's rejection in the first place of there being the dog that eat's the most.

However, the dog that eat's the least is quite understandable and with nonstandard analysis, we have even an equivalent number. So the question is open here in my view.

It is my belief, also, that although both groups are called democracies, group 2 may behave much better in cases of hardship (like natural disaster, poverty, war or some other crisis). Culture, identity and compassion may really play a role in these small democratic nations when they will face hardships. — Eros1982

I agree. In group 2 social cohesion and solidarity is far more easier to prevail. And usually group 2 countries are far more smaller, which makes democracy easier. Small size makes even other systems quite OK for the citizens under them, in fact monarchies like Monaco and Brunei can prevail quite well because it's totally possible for any citizen simply to meet the monarch and confide his or her problems to this. And when the tiny nation is prosperous and the monach isn't a madman, why not sustain that monarchy? Just think about how nice it would be if you have problem and you could simply get a time with the US President and he would look at what he could do to help you.

With regard now group 1, I think if the countries of this group face some kind of hardship, their people will show all kinds of negative behavior just because they were taught that civilization means living well and calling the police every time you have issues with your neighbor. From the moment you don't live well in group 1 and you cannot rely on the police, you either run away or you should watch your neighbor 24 hours a day. — Eros1982

It surely is a thing of simple size matters. Yet there are real differences with cultures and how they approach the idea of the collective and what's the role of the individual towards the nation. The US is highly individualistic and basically doesn't trust it's own government as much as in some other countries. In the US people have guns to protect themselves from criminals (basically other Americans) and value this gun ownership as an example of their freedoms. In Switzerland and in Finland they have a lot of guns too, but in both countries the guns aren't for protecting your home, but for hunting and protecting the state. It's just one example, but the difference is notable because it comes to other things than just the size of the country:



And it's telling that the above documentary gets a lot of flak in the US. But this was just one example how states differ from each other.

This experience of detesting contemporary American movies makes me ask the same question all the time: why in the hell people in other countries spend so much money and energies in order to see, advertise and idolize (contemporary) American cinema?

The only logical answer I come up with is "mass control". — Eros1982

Well, another reason is that making movies is actually very expensive. If you make a movie in Finnish, basically there's only +5 million people who understand Finnish. If it's a very good movie, some foreigners will see it, but not many. Think about it like Minnesotan's making movies for only Minnesotans to watch, with Minnesotans speaking a totally different language from other Americans. This is the reason why English dominates and why even the Hollywood studios themselves have centered on making "Blockbusters" and only make few "Art Films" that require a bit more to follow than just eat your popcorn.

US culture industry has a big leverage on the rest of the world. — Eros1982

Let's start from some facts: There are so goddam many of Americans compared to any other Western people. And not only that, but your are very wealthy consumers. Thus you are the biggest domestic market there is. And this means that many talented foreign directors and actors are very welcome to work in Hollywood, just as many scientists and successful entrepreneurs (like Elon Musk etc) come to the US, because the US has the resources.

Then you speak English, which was spoken thanks to the British Empire in a lot of other places. (Now if people in the US would talk not English, but French or Spanish, then either of those two be easily the lingua universalis of the World.)

In conclusion, I tend to believe that materialism and policing may have a greater saying in our modern western world than "the global culture" which I see it as being imposed on us (and easily replaceable). — Eros1982

The US surely polices competition when it comes to it's strategic interests. And my father in his time joked about the American legal battle against NIH-products (NIH meaning "Not Invented Here"). Yet all of this is actually quite limited, when tariff barriers don't exist. Especially in Latin America there is this idea of this nearly omnipotent US guarding everything in it's interests, but it isn't so. Not all largest companies in the World are American in every sector. Just take for example forestry and paper companies. You would assume just by thinking where the large forests are and think about the sizes of the countries, it would be that American, Canadian and Russian companies would be the largest. Close, but that isn't the picture, in 2022 by revenue the list was as follows.

1. Oji Paper Company (Japanese)
2. Stora Enso (Finnish)
3. West Fraser Timber Co (Canada)
4. Weyerhauser Company (United States)
5. Universal Forest Products (United States)
6. Masco (United States)

The largest US company is only on 4th place and for many it would be surprising that the largest are a Japanese and a Finnish company, which are very small in size compared to Canada and the US. But this is how globalization works. You'll find that in many sectors there are large companies that aren't American.

I can't imagine a scenario with economies and surveillance performing very poorly and with people in USA or France being in "peace" due to their "democratic/egalitarian/cosmopolitan" values and "compassion". Till, I can imagine that scenario as plausible for some smaller nations which have been lucky enough to not look like France or USA today (though I guess there must be only a handful of such nations in the western world). — Eros1982

Not quite sure what you mean here. Well, many countries don't look like the US. But what is surprising is just how similar to the US the whole of Latin America is. You have these interesting subtle differences between American countries and European countries.

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. In the story it would be simply that since every dog eats more or less than other dogs, they can be put into an order of dog1<dog2<dog3<dog4. Of course, from this we get to interesting challenge that the Axiom of Choice gives to mathematics.

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

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

Notice that π isn't constructible, but the square root of two is if irrational, is not transcendental.

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. Because if we can put π exactly on the number line, the I would argue that you can put Zeno's least eating dog exactly on the number line too. Real numbers are constructed by either Dedekind cuts or Cauchy sequences. Both use systems of going closer and closer, which simply begs there to be Zeno's dogs. In a way, with real numbers you have a lot more dogs that basically have a lot of similarities to Zeno's dogs, so much that they could be argued to be Zeno's dogs.

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

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).

And notice that the debate about just what we do accept as numbers (or mathematics) has continued and hasn't faded away. For example the Ancient Greeks didn't view like us rational or irrational numbers as being numbers: for them there were numbers and then the idea of ratios. What is accepted and what is not continues with Finitism even today, as the Cantorian set theory does still give rise to opposing arguments (especially of larger and larger infinities), even if a they are views of the minority.

For example if we want have the ability to measure the food amounts, just look at the following Venn-diagram and notice at how limited "constructible lengths" is in the diagram. 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.

What do you think yourself then? (Or if you have already given a satisfying view, please refer on what page you did it.)

It should be totally evident to everybody that when discussing the foundations of mathematics, philosophy is unavoidable. You simply cannot "just stick to the math" and not take a philosophical stance in my view.

Hence this thread is totally fitting for a philosophy forum.

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

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. — Ludwig V

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. — Ludwig V

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.

Empiricism (as embodied in the principle of testability) is just a temporary stopgap solution in science. What they really want, is the complete axiomatized theory of the physical universe. So, what they really want, is provability:

- - -

At this level, science and mathematics will be merged into one. They actually want to get rid of empiricism and testing and science as we know it today. However, in absence of the ToE, they simply cannot. — Tarskian

What if the positivist are indeed partly right, but they won't get the answer they would want to hear? Hasn't this been obvious starting from Hilbert? He got answer, but not those one's he wanted to hear.

What if this merging of science and mathematics can happen, yet not in the way mathematicians or especially positivists want it to happen? What if a lot of science and even something as distant as the social sciences is indeed mathematical, but in the part of math that is not provable or computable?

Just make this thought experiment: What if an area of study of reality is indeed mathematical, but firmly in the non-computable and non-provable, but perhaps in the "true and expressible" (as Yanofsky put it in the text that you referred in the OP)? How will this show itself?

In my view, one thing would be certain: those people studying that part of reality and it's phenomena aren't computing data or making functions or other mathematical models about reality. They will just smile if you ask if they could explain the phenomena they are investigating by forming a mathematical model of the phenomena.

Do you know about the democratic peace theory? — Linkey

Yes, but I don't unfortunately believe it.

United Kingdom declared war to Finland in December 5th 1941. I assume the both countries were then democracies even back then. (And do notice that the US never declared war to Finland, it only severed diplomatic ties as late as 30th July 1944, only few months before Finland declared war on it's de-facto ally Germany.)

And republics in Latin America have gone to war with each other, latest being the Cenepa war in 1995. And basically both Pakistan and India have been democracies, even if Pakistan has had it's share of military rule. Hence I would argue that being democracies lowers the risk of war between countries, but it doesn't erase the possibility.

Democratic countries unite instead of dissipating, and the people in the West must try to make the Russians know about that. — Linkey

Well, sorry, democracies seem far more weaker and undetermined than they actually are.

And I would urge that this is something that Russians themselves have to do. You already have had a proto-democracy in your history in the state of Novgorod, so you could easily built on that and finally overthrow the idea that Russia needs a Tzar or otherwise it collapses, which I view as nonsense.

as I have suggested, the US should declare that they will build military bases on Taiwan unless a referendum is performed in PRC with a suggestion to unban youtube. I think this is really a strong idea: as far as I know, many people in China (probably most) don't like the censorship in their country and the social credit system. — Linkey

Unfortunately those actions would only consolidate the position of the Chinese communist party and it's supporters. There would be many in the West who would see this as an imperialist attack on China and reckless warmongering.

Sorry, but the only ones that truly can liberate the Russians are the Russians themselves and so it is for the Chinese too.

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.