Russian rhetoric and behavior has been surprisingly consistent over the course of more than a decade when it comes to this issue. — Tzeentch

Was denazification on the table in 2014? But I agree, Russia has been quite consistent in attempting to annex Ukrainian territory irrelevant of NATO. As it was an "artificial" country.

They repeatedly give NATO chances for dialogue, and NATO repeatedly ignores them. — Tzeentch

Have you ever noticed what kind of dialogue that was? It was that Russia should have a say if a country could join or not NATO. That naturally goes against NATO's charter. At least for Finland that was the second to last straw to break (the last straw being the full invasion of Ukraine).

And of course, what you always forget, is that simply putting the troops on the border made immediately NATO countries like Germany vow that Ukraine wouldn't be accepted to be in NATO. But naturally that wasn't the goal, just as Saddam's mobilization of Iraqi troops to the Kuwaiti border wasn't done in order that Kuwait would follow OPEC guidelines on oil production.

Where is this imperialist Russia that wants to "Finlandize Europe"? — Tzeentch

Here's Russian foreign minister Sergei Lavrov, from the official Russian Foreign ministry website :

As for the “Finlandisation” of Europe, I remember that period very well. It was an element of euphoria that developed after the end of the Cold War, when everyone was considered a friend, and ideology was abandoned everywhere.

https://mid.ru/en/foreign_policy/en/foreign_policy/un/1959636/

That's the delusional way that official Russia thinks about Finlandization.

In fact, the Soviet Union hoped the Finlandization would catch on. It's simply how it wants to influence other countries. The best way is to "influence" the actions of other countries without having an open conflict. The worst thing for this idea (and the present Russia) is European countries forming an union and the Atlanticism that NATO creates.

But we need multiplication for Godel numbering. — TonesInDeepFreeze

OK, I think you answered here my question.

Had the West not insisted on changing Ukraine's neutral status, Russia probably would have never invaded. — Tzeentch

This is pure "what if" arguments, which are unprobable and now .

As noted far earlier in this thread, there were aspirations for annexing Crimea right from the start when the Soviet Union collapsed. Then there's all the talk, all the aspirations for getting Novorossiya and hence carving up Ukraine... as now has happened.

But this is totally futile debate as we simple here disagree. You insist that everything happened because of NATO enlargement and the West.

We have been over this, so no reason to go again in rounds.

We know it is so because having both addition and multiplication entails incompleteness, so, since Presburger arithmetic is complete, it can't define multiplication. — TonesInDeepFreeze

And that was my basic question: why having both addition and multiplication entail incompleteness?
How does it entail incompleteness?

Is it that with both addition and multiplication you can make a diagonalization or what is the reason?

Indeed that's interesting. With Robinson arithmetic you rule out mathematic induction and the axiom schema. But you do have the successor function, addition and multiplication...and that seems to be all it takes for incompleteness.

On the other hand, for instance Presburger arithmetic is complete. But then:

Presburger arithmetic is designed to be complete and decidable. Therefore, it cannot formalize concepts such as divisibility or primality, or, more generally, any number concept leading to multiplication of variables.

If anyone has something more to say about this and why this is so, I'll definitely want to hear from you.

I agree with Lichtman's argument on the power of incumbency. If there are no quagmire-conflicts going on, if the voter's 401K's are up and there is no recession, the incumbent is posed to win. And Americans seem to have forgotten that they indeed lost a war in Afghanistan as nobody is talking about it, since it was also Trump's fault also. And yes, there's no billionaire third party candidate running.

And I agree with Lichtman that Democrats are shooting them into the foot, but then again no incumbent running for the second term has been ever so senile (at least in the open). So even Lichtman is unsure of the outcome and waits for the Democratic convention.

However, even if Trump wants to follow the Orban model, as Lichtman says, what this great populist orator lacks is the needed leadership qualities. We already know this from the last time. And the idea that Trump can wreck American democracy, well, just look what happened on January 6th:

Trump wanted to go with his supporters to Capitol Hill, but his own security team simply drove him to the White House. There he watched mesmerized from television how his supporters took over the Capitol Hill building and finally, after calls from his inner circle and family, he just said to his supporters to go home.

Now, sincerely, ask yourself: is a person that acts in this way even capable of overthrowing one of the oldest democracies in the World, if he actually wanted to do it? Because you don't get a better chance ever for an autocoup like January 6th, with your supporters breaking into Capitol Hill. Democrats were totally stopped in their tracks as a deer in the headlights on that day. But then you would have to have a real plan, you would have to have people that support you, understand it's either they go through all the way or they face a life sentences, even capitol punishment. Nothing like that happened.

Then ask yourself: is now the Republican party really intent on wrecking democracy? All of them?

Or does JD Vance, a former marine that wrote in 2016 "Mr. Trump Is Unfit For Our Nation’s Highest Office" among other comments, wants to now wreck democracy of the US? Isn't the last VP of Trump a clear example of the Trump team not having these kind of thoughts?

If you think that the intent of Republicans is this, I disagree with you. I think you taken in too much of the rhetoric which causes the political polarization in the US.

Hence in my view in 2028, even after a Trump presidency, there will be a democracy in the US. What kind of ride would it be to 2028 is a different question.

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