If the knowledge or insight is worth money, yes. However there's still a lot of academic and scientific studies that people, who have done them, would enjoy if their ideas would be picked up by others. It's totally understandable that for example nuclear weapons technology and the science used is declared classified and you just don't sell it to anyone that wants it. With philosophy, it isn't.

Still, I think that there is a problem when there simply are so many scientists and academic researchers, group behavior kicks in and an incentive emerges to create your own "niche" by niche construction: a group creates it's own vocabulary and own scientific jargon, which isn't open to someone that hasn't studied the area. Then these people refer to each others studies and create their own field. Another name for this could be simply specialization: you create your own area of expertize by specialization on a narrower field. When there are a masses of people doing research, this is the easy way to get to those "new" findings. Hence even people in the natural sciences can have difficulties in understanding each other, let alone then the people who are studying the human sciences. Perhaps it's simply about numbers: 30 scientists can discuss and read each others research and have a great change of ideas, but 3 000 or 30 000 cannot. Some kind of pecking order has to be created. The end result is that you do get a science that is "Kuhnian" just by the simple fact that so many people are in science.

(In 1927 it was easier just with 29 quantum physicists meeting each other)


I remember when studying a course in philosophy in the university I personally got irritated when a philosophy teacher said that one could only use the German words for some philosophical terms or otherwise their "essence" would be lost (think example's like Heidegger's Dasein). Well, if you cannot translate your ideas to another language or cannot define something by using other words, there's something wrong with you. But that's just my opinion and others can disagree with this.

And this is universal and not just limited to philosophy. I remember another historian who went to great lengths to write one of her historical books to be as easily readable for the layman as she could do only then to be scolded by her peers for the book not being "academic" enough. For some to be as understandable as possible isn't the objective, the objective is to limit those who don't know the proper terms out of the discussion, even if they could participate in the discussion. Naturally people will simply argue that just like with abbreviations, we make it easier for people to read it when we use the academic jargon. But there can really be other intensions also.

(Yes, it's the length of the equation, even if mathematical beauty would say otherwise)

As a starting place maybe it'd be nice if public libraries had access to academic journals. Taxes go to pay for that research after all. It should be accessible. — Moliere

Better even if academic journals would be available freely in the net. Think about general science magazines if they would have links to all the original publications. There's still the paywall and simply you have to somehow have to have the knowledge of an interesting article being in some publication.

I'm interested in something called arbitrary transfers. Basically any transfer of money without any goods or services in return. — Mark Nyquist

What is arbitrary about them?

Even the US government is spending about half into transfer payments to individuals and businesses (mostly to individuals). This include Social Security, unemployment insurances, student grants, etc.

Welfare states do it even more.

Something along those lines has come up before. — jorndoe

As many other things and issues. Over and over again in this thread.

Russia has a population of 144 million, and a GDP of 2.2 trillion.

It's tiny. Germany alone doubles Russia's GDP. — Tzeentch

Doesn't matter when Germany has a tiny army that could be swept away. Just look at the difficulties that Germany has had in increasing it's defense spending. Bureaucratic Germany with it's people not threatened with Russia isn't going to budge easily.

War with the Russians is in European interests?

The fact that you're saying this is why I keep emphasizing you're war-hungry to the point of absurdity, and you don't even seem to notice it yourself. — Tzeentch

You have to have deterrence to keep the peace.

Now creating that deterrence will simply get some people to think that your war-hungry. Well, I'm not.

But the cold fact is the deterrence is to prepare for war. What else would it be? You shouldn't forget that.

We are no longer a key ally to the United States, since we are nowhere near the Pacific and likely to stay on the sideline if large-scale conflict were to break out there. In fact, we will profit from a war in the Pacific, directly and indirectly. That now makes us a threat and a potential rival to the US. — Tzeentch

First of all, the globalized World won't profit from something far more devastating than a trade war. The disruption to the global trade routes and supply chains would be far more bigger than anything experienced during the Covid pandemic.

And your forgetting that the US has nothing like NATO in Far East. Don't you remember how SEATO simply collapsed? What are the goddam allies of the US? How close are South Korea and Japan to make some joint effort here? What are US allies there in the Pacific? Australia, and the UK! Not much of an alliance that AUKUS.

This is the peril when you have only nation-to-nation defense agreements, but not a treaty organization with collective defense. What countries would (or could) assist the US, if China went for Taiwan? The Japanese? How much? The South Koreans? They have to deal with North Korea. Likely Japan could give a few destroyers and subs, but likely it would hold it's resources back. And in truth the US is lousy in creating new workable alliances, because it doesn't want to.

In truth, the more the US gives the middle finger to Europe and NATO, the more it plays to the hands of the Russians and the Chinese. Just like forcing Ukraine to give up will just play into the hands of Putin and Xi Jingping. And the more likely it will want then for assistance from NATO, if there is an actual conflict in Taiwan.

Sure. But it needs to do so without pointlessly antagonizing Russia, otherwise rearmament is going to lead to mutual tensions and militarization (which we are already in the process of), which will not achieve security, but the exact opposite: war - which is of course exactly what Uncle Sam is trying to achieve in Eastern Europe. — Tzeentch

Pointlessly antagonizing Russia?

Look, Putin has all the time wanted to portray Western Europe as a threat Russia. That Russia is under threat is the reason he can claim to have to have dictatorial powers. Every opposition to his regime is a CIA backed attempt to break up Russia in his view. Fear of the West is the essential oxygen that the Kremlin breathes. I think that you simply don't understand this and all the incredible bullshit the Kremlin is really creating about the West. Just like this one, just to give one example:

(The Barents Observer, August 2023) One of Russia’s most powerful men this week paid a visit to the Republic of Karelia, the region located along Russia’s border to Finland. Nikolai Patrushev is Secretary in the Russian Security Council and a close ally of Vladimir Putin since the 1990s.

In a meeting on national security held in Petrozavodsk, the Karelian capital, Patrushev warned against terrorist attacks from the West.

“In Karelia, ten crimes of terrorist character planned by “westerners” and Ukrainian special services have been averted only over the last half year,” Patrushev said. He claims that western security services are deliberately trying to stir separatist tensions with the aim of destabilising the political situation in the region.

And the foreign services are especially working towards youth, he argues.

“On social media, separatist calls are propagated [and] western special services’s sabotage and terrorist pursuit in the region constitute a security threat to population,” the security chief said.

Patrushev’s meetings in Karelia are covered by regional government newspaper Respublika.

According to Nikolai Patrushev, additional measures must now be taken to disclose and avert terrorist activity in the region. And effort should be concentrated on transport infrastructure, as well as energy objects and objects of biological and chemical character. Also teaching institutions and public places of mass gathering of people must be given priority, he underlined.

The Republic of Karelia has close historical ties with neighbouring Finland and cross-border relations thrived through much of the 1990s and early 2000s.

(See here)

This is the kind of bullshit rhetoric coming out from the Kremlin to it's own people. And then you think creating deterrence is antagonizing Russia too much?

As though US geopolitical strategy is going to be guided in any way by what NATO countries want, as oppossed to what necessity dictates. — Tzeentch

Actually it is guided. Biden was all in favour of the "pivot" to Asia and his administration full of the "pivot people", just like Obama's. But he cannot and couldn't. That's the power of Atlanticism.

For Superpower USA, the North Atlantic Treaty Organization is the jewel in it's crown. But if American want to let their hubris go rampant and think their stature in the World doesn't need NATO or any allies (except Israel), I think they are mistaken. If the US really leaves NATO, then Europe will have to reorganize itself. But it's not something where the US wins.

Believe it or not, but I don't think that especially those Americans wanting to "make America great again" don't want the country to be the bigger Canada. Yeah, isolationism sounds great at first, but then when other countries really don't give a fuck, then comes the anger back when people find that isolationism is not a cure-all, just as Brexit wasn't for the Brits.

The simple fact is that the West is stronger together. Something that some people hate.

Incommensurability would indeed be an interesting question, perhaps even of an own thread. At least with icommensurability, the question of infinitesimal being larger than zero but then could be discarded might work.

To show that the idea of Zeno's dogs least and most eating dogs, infinity and infinitesimal as a number, isn't so far off and how non-standard analysis has made the infinitesimal a come back is explain for example in the following video:

What you're describing are US-European relations during the Cold War. During this time, Europe was a key US ally against the Soviet Union.

What I am saying is that this relationship has been fundamentally changing since the end of the Cold War, and especially since China has emerged as the new threat to US power. — Tzeentch

In Europe, especially countries like Poland, Sweden, Finland and the Baltic States see the situation as the continuation of Cold War. Hence they usually are flabbergasted when (and especially after 2022) when some idiot starts talking about the present as totally different from the Cold War.

The idea of a fundamental change is nonsense. It was the nonsense when it was eagerly talked in the 1990's, when the membership of Russia in NATO was on the table. Then it was the "New Threats" and things like conscription were "ancient relics of a bygone era". Not anymore. If you wouldn't have ex-KGB officers at helm in Russia, yes, Russia could have been totally different.

What has changed is that Germany is unified and now has a bulwark of Poland between it and Russia. For Poland the situation is far more perilous than it was in the late 1990's (and thus it's vast rearmament program).

Since the Ukraine war, it has become increasingly clear that US and European interests divert in a way that is dangerous for Europe. The US is using European naivety in this regard to have us hamstring ourselves. — Tzeentch

I have no idea what you are talking about here. European and US interests are quite the same: the totally reckless territory annexing Russia threatening with nuclear weapons is a threat. You seem to be in your own echo-chamber.

Supposedly we were going to feed Ukraine weapons to hurt the Russian military so they couldn't pull another stunt like Ukraine, yet it's the European militaries which are completely stripped and the Russians who now have an army several times the size of their peace-time standing army. — Tzeentch

That's why Europe simply needs to rearm. The assistance it has given to Ukraine has in now way been a real burden.

We are now no longer "friends," but temporary assets to the US, and the US is already preparing the ground for when Europe finally slips its orbit. — Tzeentch

Bullshit again.

NATO countries don't want the US to go. And the idea that Americans want to be just like Canadians isn't true. But that's the reality if the US wants to go out of "foreign entanglements" like NATO and Atlanticism: then the US is just a large Canada. People don't know (or care) in Europe what the Canadian government wants. But naturally everybody has quite good relations with Canada.

And why this thinking that the countries want to severe good ties? That's the thing in your anti-American hostility you forget.

There's no point in talking about the Soviet Union. Russia today has nowhere close to the power of the former Soviet Union. It's a completely different country. — Tzeentch

Again wrong. Russia has a very large armed forces and a nuclear deterrent, while European armies have fallen in size dramatically from the Cold War era. And present day Russia is as aggressive if not more aggressive than the Soviet Union.

But Finland has been working with NATO for a long time, and has been a member of the European Union almost since the start. — Tzeentch

Assuming that the EU started at 1993 is wrongful, the treaty of Rome in 1957 would be far more correct. Naturally Finland has tried to have as much of ties with the West, earlier there was naturally the obstacle of "Finlandization".

Obviously once tensions start rising as they did post-2008, Finland is going to be in the crosshairs. That's where it put itself when it aligned to a bloc that became hostile towards Russia. — Tzeentch

You really don't get it, do you.

There's no other option. There's no option of "Let's be friends with Russia" that would have a better outcome for Finland: Russia would just increase it's efforts to dominate Finnish policy, if it would be let to do it. With Russia as a neighbor there's no "normal" relations let's say Canada and Switerland can have. Just look at how great being the member of CSTO has been to Armenia. Russia didn't give a fuck when Azeris supported by Turkey annexed Nagorno-Karabakh. That's the kind of "ally" Russia is. CSTO, just like the Warsaw Pact is primary a form of control.

I think in your naivety you think that if only Ukraine given up on everything and done as Russia wanted, everything would be fine. It wouldn't. Just read what Putin wants. Ukraine was an artificial construction and the country's should be together. That's it. All those annexations show that this isn't just about NATO and Americans, it's about a lot else that you simply refuse to take into account.

You are quite selective with the lessons you learn from history, I've noticed. — Tzeentch

At least I refer to the peace agreements and the wars the Russia / Soviet Union has fought, unlike you.

Do we learn nothing from Vietnam, the Middle-East, etc. when it comes to US involvement in fragile states? — Tzeentch

Fragile states like North-Vietnam or Israel? Hmm...

Or you really think that Ukraine or Finland, Sweden, is similar to South-Vietnam or Pro-Western Afghan government?

I think what plays a large role is that, despite all the historical evidence, Europe seems chronically incapable to view the United States as a ruthless great power which follows realist logic. — Tzeentch

This shows how you really don't understand Europe. You think that US and Russia act and behave in Europe similarly, because they are Great Powers.

I assume that you come to this conclusion with thinking about how the US has treated let's say Guatemala (and how the US has acted in it's backyard). Well. in the long run the US policy towards Guatemala has been more like the United Fruit Company's policy towards the country. The US doesn't behave similarly towards France, Sweden or Finland (as Russia doesn't behave similarly towards Brazil and India as it does towards Georgia, Moldavia or Ukraine).

And the simply fact is that you simply don't seem to understand European integration and NATO at all. NATO isn't like Warsaw Pact, which primary function was seen in Hungary in 1956 and in Czechoslovakia in 1968. NATO isn't just a puppet for the American President, which has been shown quite many times (for example from how much Trump despises the organization this should be evident). You insist Ukraine would have been invited to NATO, because American presidents wanted so, even if it was obvious that many NATO countries opposed this. Yet NATO could never give formally an outside member. And the de facto assurances didn't matter for Russia, because it has territorial interests in Ukraine. Just like EU hasn't officially stated that Turkey cannot be never an EU member, this is de facto real. It's an international defense pact that sovereign states have willingly put their defense into, just as EU countries are committed to European integration.

The simple fact is that even if for Yemenis or Palestinians and many Latin Americans, the US seems to be a ruthless Superpower, but that isn't the case for Sweden, Finland or East European countries. Just as Russia wouldn't never dare to do any hybrid attacks towards India and try to involve itself in Indian politics. I'm sure Russia behaves quite cordially towards it's BRICS partners. It doesn't act the same way in it's "near abroad" thanks to being and seeing itself something else than a nation-state, but a great power. This is something you have to understand, but you just ignore it.

So Finland and Sweden gave up their neutral status and put themselves in the crosshairs of a future conflict to 'protect' against a power that was trying to return to stability to begin with. The power who is trying to avoid a return to stability is the one they chose to jump in bed with. — Tzeentch

I do value your opinions, Tzeentch, but this is ignorant bullshit.

Especially Finland has been in the crosshairs of a conflict with the Soviet Union starting from the armstice in 1944. It was in the crosshairs and continued to be in the crosshairs especially after Putin has wanted to make Russia a Superpower again. Russia did it's hybrid attacks by organizing refugee flows into Northern Finland in 2015-2016. It has GRU sleeper cells in the country ready to do sabotage and to assassinate important people as the way of it's "deterrance" in Finland, if war breaks out. It has breached consistently Finnish aerospace with military aircraft, has jammed GPS signals and kept up belligerent rhetoric all this time prior to 2022, hence Finland has all the time been in it's crosshairs. What you are saying is simply ludicrous.

You simply don't understand that there wouldn't have been any end to this if Finland would have stayed neutral, likely the hybrid attacks would have continued even more to push Finland back into a weaker spot. There would be no "normal relations", there would be only Finlandization, where the Finnish President would get his international speeches from the FSB chief in the Russian embassy. That's the fucking "normal relations" that Putin wants. That we would talk the "lithurgy" as in the Soviet times.

And wtf stability are you talking about? Russia's military has always been multiple times larger than Finland's or Sweden's? Do you think annexing territory from Georgia and Ukraine is a way of Russia attempting "stability"?

It's simply imperialist revanchism, an attempt to fix what Putin sees as the greatest tragedy of the 21st Century (collapse of the Soviet Union). Nothing else.

You clearly have a problem with the idea that things can return to normal after this war, even though it would likely be the best scenario for all parties involved (except the US). Why? — Tzeentch

I don't.

The war ends when the two sides come to some conclusion, either a peace deal or a cold armstice (as with the Koreas). And it's the job of Ukraine and Russia in the end. And I don't want to hear your arguments of Ukrainians being pawns with no ideas of their own in this game.

The thing we've learned from history is that Russia has to be forced someway to a peace agreement: if continuing the war looks to be a better option, the Russia will continue the war. Plain and simple.

And likely relations are tried to be improved after Putin dies of naturally causes or is replaced. Remember that the war isn't hurting Europe so much, so this can go on for years, even a decade.

Second order recursion sounds indeed more logical.

What do you mean by 'recursion of the reals'? Recursion requires well ordering. — TonesInDeepFreeze

From the quote, the only difference between recursion and real recursion, that I can think of, is using recursion for reals, in other words recursive real numbers: "A recursive real number may be described intuitively as one for which we can effectively generate as long a decimal expansion as we wish, or equivalently, to which we can effectively find as close a rational approximation as we wish."

Because I don't think there's real recursion and "phony" unreal recursion.

Thanks for the comments!

One point though: Godel-numbering is in the meta-theory, but we want to know why we need multiplication in the object theory. But, if I'm not mistaken, we need that it is representable in the object theory; I'd have to study the proof again. — TonesInDeepFreeze

Yes, I understand that it's a part you need in Gödel-numbering, to make the number that holds the logical sentence. Once you have both addition and multiplication, you can do what Gödel did. With Presburger Arithmetic the completeness is lost if you take into account also multiplication:

adding, for example, the multiplication function x: Z^2 -> Z(or even simply the ‘squaring’ function, from which multiplication is easily recovered) to Presburger arithmetic immediately results in undecidability, thanks to Gödel’s incompleteness theorem

(see On the Decidability ofPresburger Arithmetic Expanded with Powers)

But then I also found an interesting answer on StackExchange, which seems very interesting, an answer to the question "Why does multiplication lead to incompleteness where addition does not?":

The issue is not about multiplication per se, or even about the combination of addition and multiplication. The theory of Real Closed Fields has both, and is consistent and complete. The issue is about the strength of induction.

The induction axioms in Presburger arithmetic are the first order approximation of the Peano axiom, and basically do not allow for establishing facts about other facts that have, themselves, to be established inductively.

You cannot get real recursion off the ground unless you have a second order theory of counting, that allows you to represent the sets of integers for with the results are already established.

So to get a first order theory to start doing Gödel's proof, you have to bring in either infinitely many facts about addition, which are needed to establish the relevant results about multiplication, or a few facts about multiplication, itself, as additional axioms.

(See here)

I find this interesting. With real recursion I assume means recursion of real numbers. Well, if you have real numbers, then you're hopelessly mired in the notion of infinity and infinite sequences and so on. With mathematical induction, we get to questions about infinity.

Ukraine's neutral status is the key to a stable Eastern Europe. — Tzeentch

Everybody even Ukraine would have been totally happy with Ukraine being neutral... assuming that Russia wouldn't have intension of annexing large parts of Ukraine into itself, as it has done. And this simply is the reason for all of this. I think you have a problem in understanding just how a big deal is it to other sovereign states for a state to attack another one (which it has earlier recognized) and then to annex parts of it.

And this is the reason why Russia itself creates the situation where other countries want to join NATO.

Just stop and think it yourself for a moment: why would Sweden with a leftist government want to shed it's over 200 year neutrality and Finland, that earlier enjoyed the fruits of having good relations with Soviet Union and later Russia, suddenly join NATO? You think it was an American plan?

It's worrying how your rhetoric turns any dialogue with the Russians into something negative. — Tzeentch

The majority of Putin's rhetoric is negative. Not all.

Just like the way you use the term 'Finlandization' to describe any kind of positive relations with the Russians. — Tzeentch

WTF are you talking about? We had good relations with Russia. Finlandization has a negative definition, which as a Finn I clearly understand.

Finlandization the process by which one powerful country makes a smaller neighboring country refrain from opposing the former's foreign policy rules, while allowing it to keep its nominal independence and its own political system.

But your stance is that if a country attacks another and starts annexing parts of that country (and actually has done this to two of it's neighbors), then other countries should continue to have perfectly normal relations with this country. And if countries oppose the aggression by severing ties and sanctions, that's bad. Oh, that's so bad!

It bears every hallmark of war propaganda, which is designed to make war the only outcome. The same trick was used in Ukraine to make it fling itself willingly into the abyss. — Tzeentch

Blaming the victim is so nice. :vomit:

The question you should ask yourself is whether you will be the beneficiary of such a war, or whether that will be some unnamed country across the pond. — Tzeentch

I'm a great supporter of deterrence: with good deterrence, you can avoid blackmail and war. Without any deterrence, Great Powers will do as they want with you. Russia is and has been this kind of Great Power that if it see's an opportunity it will use it, especially in it's former "colonies", even if we don't talk as colonies. Well, my grandparents were born in a Grand Dutchy of Russia. That was basically something similar to France in Algeria. And Russia views it's "near abroad" as similar as other Great Power viewed their colonies. With Putin at the helm, Russia hasn't moved on from it's imperial past and simply continues similar policies as earlier and views it's near abroad as it's own. Unfortunately, it isn't as benign as the US is to it's neighbors (at least after it had it's wars with Mexicon and the British Empire). And this is why other countries like mine that were for a long time non-aligned have chosen NATO. Annexing territories is the real key here.

I would be really happy if Russia would be a normal country and respected the borders of it's neighbors. I have nothing against Russians, I have known many, I have been in Russia and love it's food and culture. But it is a dictatorship that lost in it's gamble and is mired in a war.

Your the one talking about enlarging the war, not me.

* * *

And from an earlier response, I'd like to add:

Russia has a fraction of Europe's GDP and population. Russia is hardly a threat if the Europeans would just get their heads out of their asses. There's no basis for this type of fearmongering nonsense. — Tzeentch

You simply don't even read what I write: Russia with it's large armed forces and with it's huge stockpile of nuclear weapons is more than a match against any EU country vis-a-vis. And with the US out of the equation, the military balance is quite on the side of Russia even if you group up European countries. And then there's just all the hybrid operations that Russia has done, which you dismiss, of course.

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.