There was a similar argument some years ago about the Falkland Islands, that every single local could be bought off by Argentina. I think most Americans are much more mobile and not used to the idea that someone could be strongly attached to an area as their home.
Canadian perspective: it seems like many (not all) Americans are also explicitly brought up with, and heavily absorb, "greatest country in the world" rhetoric that also leaves them incapable of understanding that this is not necessarily how people outside of it see the USA.
Trying to be diplomatic? I've seen a lot of really insane shit in comment sections -- and even more insane shit from public officials -- over the last year since these "topics" came up.
Belief in or unconscious absorbance of the concept of "manifest destiny" and American exceptionalism seems to run deep in a portion of the American psyche.
No, it's not "their" continent to take. Might doesn't make right. And it certainly doesn't make for superiority in anything but brawn and bravado.
I'll also add this: the (imho encouraging) trend in the north and with the indigenous populations there has been in the opposite direction from what the Trump regime is proposing: sovereignty and autonomy for the Inuit to run their own lands. Canada carved out new Inuit administered territories, trying in some respect (and inconsistently) to rectify over a century of mis(mal?)governance and exploitation and mistreatment.
From the polls I see the Inuit in Greenland want more self governance, not a new external boss.
From an American perspective, it’s not the “rhetoric,” it’s just “noticing.” My mom is an immigrant who was not brought up here to absorb the rhetoric. But when she went to Canada and Australia to visit family, she came back ranting about how poor everyone was and how small the houses were. (I take it you have fewer suburban McMansions and giant SUVs.) It’s hard not to notice our GDP per capita is 50% higher than yours. It’s big enough now where we notice it just going up there to visit family.
And you can say what you want about safety nets for poor people, but that doesn’t affect most Americans. My parents are on Medicare and they head down to the ER every time have a stomach ache and get a CAT scan. Meanwhile my family is convinced that Canadian healthcare nearly killed my aunt when she had a kidney issue because they didn’t immediately schedule her for a million tests and surgery. (I suspect that isn’t true and the Canadian system reasonably triaged the care.)
And to be clear, I like Canada (and I love Denmark). I’d rather have a more orderly society with an efficient and expansive government that’s focused on comprehensive outcomes across the population, in contrast to our system where you have McMansions but randomly you can fall through a giant crack. But Americans temperamentally are biased towards upside potential and they devalue downside risk. This is a cultural trait that seems very quickly absorbed even by immigrants. My immigrant family isn’t meaningfully American in many respects—they don’t have Anglo sensibilities about things like civic institutions and personal freedoms—but they’re indistinguishable from other americans in their materialistic optimism
> From an American perspective, it’s not the “rhetoric,” it’s just “noticing.”
Yes, but that does not go against the parent comment. When you grow up in it, you have it in you, and it's difficult to question it. If you ask Americans who live abroad, they often have a more nuanced perspective.
Apart from American finding it better in the US than everywhere else in the world (your "noticing"), there is this tendency from Americans to genuinely believe that the rest of the world agrees with that. "Everybody wants to live in the US because it is the best country in the world".
And this is very, very far from true. It's not just about money. The US have a lot of fossil energy, which is good for their economy, which is good for their military. The US is a big and rich country, which makes it powerful. But that is bad for the countries and people who are threatened by the US (and recently the US have been militarily threatening countries who until then were seeing the US as an ally or at least a friend), and it is bad for our survival (through the climate and biodiversity issues).
Tons of people outside of the US wouldn't want to live in the US, even if it meant earning more money. And on top of that, tons of people outside of the US feel threatened by the US, for good reasons.
This comment underscores how mono-dimensional some people are.
To Rayiner more stuff, bigger stuff = happier and more fullfilling life. An incredible lack of depth.
That is also the reason why Americans when they go abroad are astonished and always come back saying "people are amazing" somewhere else, well no wonder considering the state of domestic affairs and domestic relationships between people.
Please offer my apologies to your mom , as it's true that our vehicles are dangerously underdimensioned , maybe next time something in the order of 10-15 short tons could be adequate to transport her to the nearest McMansion (or McDonald's rather).
A more diplomatic way to say it would be that it is a different culture. And I would agree that Americans struggle to see that other countries have different cultures and different priorities.
If you believe that the goal in life is to live like an American, then obviously the best at doing that are... the Americans. The mistake is to not recognise that other people may have different beliefs.
> To Rayiner more stuff, bigger stuff = happier and more fullfilling life. An incredible lack of depth
Please read my whole post! I’m a Europoor at heart. I live in a 3BR house with three kids and no yard despite being able to afford a bigger one. I drive an EV, and it’s not a Tesla. I’m just trying to convey my impression of American culture though the lens of my mom, who embodies this aspect of American culture quite strongly.
Yo, chill. You can always make your point just fine without resorting to aggression or making it personal.
I’d suggest treating this as an opportunity to reflect a bit on patience and tone. Speaking from firsthand experience, these are things I’ve had to work on myself over time, helped in no small part by this community and @dang’s exceptional patience for which I'm quite grateful. I now owe HN therapy money as well, lol. :)
More generally, it would be nice if we made a conscious effort to keep HN a little less negative overall [1].
I honestly can't see a difference in housing sizes between Canada and the United States -- we've got the same McMansion sprawl all over the place here -- so strikes me this person's mother is just like every other human, and bad at statistics and handling their own biases.
The US does have a higher rate of wealth inequality than Oz and here in Canuckistan tho.
Yes, that's why I found 'house size' a strange complaint. The three countries in question have comparable square footage - the largest-sized houses in the world.
The inequality is what she’s reacting to. Most people in my extended family are professionals or business owners. That class has a lot more money in the US. Top 1% in Canada is $315,000 while in the U.S. that’s outside the top 3%.
Yes, it's absolutely the case that people in our profession and adjacent do a lot better in the US than here.
And the situation for working class Canadians isn't great either right now -- housing prices have skyrocketed. Tariffs from economic warfare are destroying the labour market. There are many aspects about our situation that are inferior.
But guess what -- that has fuck-all to do with how we perceive the relative value of our country or the pride or love we have for our homeland and love.
No, the majority of Canadians don't see the US as the world's best country because the wealthiest there make more money than the wealthiest here.
I worked at Google in Waterloo for 10 years. At any point I could have packed up and moved to the Valley and transferred to Mountain View. I had jobs before that that could have taken me to the US on transfer, as well. I chose not to. Why?
During part of that time, after Trump was first elected, I saw lots of expat Canadians who had been working for Google in the US return and transfer back to our office. They came back and earned less, and the choice of projects in our office was slimmer. But they chose to. Why?
Love of country, of culture, of family, of nature, of the land, nostalgia, familiarity. What came up often when I spoke to people coming back was a strong distaste for the idea of bringing their children up in the American education system with its extreme degrees of inequality, status seeking, elitism around "Ivy League" and ranking of schools right from kindergarten. Values on the whole unfamiliar to the same degree among Canadians.
Expats in particular, and immigrants who primarily migrated for economic reasons... yes, I'd naturally expect them not to understand this POV. I even meet plenty of new (often temporary) Canadians using Canada as a convenient springboard before their "final" migration choice which is the US. Not sure I like that, but that's their choice.
My father is also an immigrant, from Germany. He came here for the nature / wilderness. He's intensely critical of the politics and economics here and where he lives in Alberta, and there's many things in those respects he prefers about Germany. But he has love of land, and Canada is his homeland, because of the peace and love he finds in the rivers, the forests, the muskeg.
I love my people and country I imagine in the same way or similar way Greenlanders love theirs. The size of the McMansions has no bearing on it. Canadians by and large don't walk around proclaiming theirs the best country in the world. We are not interested in our superiority. But we will defend our home, same as any other.
The original point stands -- to talk about "greatest country on earth" and then act baffled or smug about why others wouldn't want to join it -- is nothing but schoolyard bully logic. Like picking on the weird or weak kid in the playground, and then proclaiming that as a moral virtue. This Greenland stuff, and the rhetoric heard about Canada this past year as well, has exposed the very darkest underbelly of the US. One we have seen here many times over 200 years, but many Americans seem blind to.
I know some that did move to the US for economic reasons only that have moved back to Canada because of the way the US has changed during their time there.
I take a drive through Detroit and I "notice" entirely different things which somehow your screed above is mostly blind to.
That's about as diplomatic as I can summon up as a reply to your comment, whose substance mostly proves my point about the bizarre exceptionalist world Americans seem to occupy in their heads. It really isn't "noticing"... what you're talking about. It's ideology.
Also GDP per capita is the kind of garbage metric I would expect someone frequent on this forum, and hopefully literate in statistics, to understand the ridiculousness of deploying in conversation.
Also, there's rarely anybody more invested in seeing the superiority of their new (chosen) place other than immigrants, so I don't think that's the argumentative flex you think it is.
As I said, I like Canada! I’m just trying to explain the American point of view. For example, I care about Detroit. But your typical American doesn’t live in Detroit. The average new home is built in a booming, low tax, Sun Belt state like Georgia and Texas, where my cousins bought McMansions in the last few years.
Also, my cousin grew up in Windsor and having been there plenty of times, it’s shit too.
Much of the world would have no problem with americans being in love with their McMansions.
But many would find them wasteful, and a terrible place to live, compared to a decently sized apartment (one 10m² - 100 sq ft - bedroom per person/couple and maybe an extra office) in a walkable town.
Just as we find american SUVs totally inadequate compared to our cars.
And it's not matter of cost, we're perfectly “happy” paying borrowing millions of euro for such apartments, and paying far more for our cars.
> This is a cultural trait that seems very quickly absorbed even by immigrants.
Sure, for those who stay. If they didn't like the American way, they probably wouldn't stay. I have known many, who have stayed in the US for exchange, expat and science. They all had a great experiences, but no one wanted to stay as immigrants.
Ice is inconsequential. They enforce immigration law and Europe is way stricter than the US. It's also much harder to integrate as European nations are ethno states (no hate.... Nothing wrong with it. Most countries are. Just a statement of fact)
> I am in my 50s, I have never seen a country as divided and toxic as the US, and that is for a reason.
So you lived from 1970-2020, literally the most peaceful time on the planet due to Pax Americana. If I were a European nation, I'd just give the US what it wants. Especially maybe try not to appear like they just want to desperately hold on to their colonies? I mean, where's the decolonization people when you need them
> Especially maybe try not to appear like they just want to desperately hold on to their colonies? I mean, where's the decolonization people when you need them
Greenland’s path to independence is clear in law: only the people of Greenland can decide, by referendum. Denmark would respect that choice. So far, Greenland has not chosen to start the process. The decision is entirely Greenland’s.
My perspective is from Scotland, and what annoys me is the vanilla press continually going to the Copenhagen government and even the Danish royal family for quotes. Greenland is a Danish colony/overseas territory (delete according to view), and I'd prefer to hear from Greenlanders. I'm sure the average Dane rarely thinks about Greenland, or didn't until recently. (Much like most British never thought about the Falkland Islands until Argentina invaded.)
We had a similar scenario during the Scottish independence referendum with the international media going to London to talk about the matter.
As for the Manifest Destiny thing, maybe I'm wrong here, but I'm thinking more Monroe Doctrine. Manifest Destiny was heading westward, and grabbing Greenland, or even Venezuela, seems more aimed at those who would influence them from outside the Americas.
Monroe Doctrine I think has always been considered to apply to Latin America. But probably mainly because the US has almost always (in the 20th century) had a subservient "partner" here in Canada.
But we were invaded (twice) here in Canada (or what became Canada) by the US. The only people to have ever invaded us. And when they did so, their leaders at the time were definitely flying the Manifest Destiny rhetoric. So much so they could not even imagine why the Quebecois and others didn't just welcome them with open arms.
So, no, I think it definitely applies northward too, not just westward. Or at least some of the ideological underpinnings of it.
Greenland, and Canada are in the Americas, as are Venezuela and Cuba. Three of these are independent nations while one is more or less a European colony. Venezuela and Cuba have strong ties with Russia that the USA resents.
Yes, I'm aware of the US attempts on Canada. I think US pop culture, and TV has done a better job of Americanising Canada than the military... Same with Europe.
I tend to think of it more like this: There is a North American culture and then subcultures. "Canada" is in large part contiguous culturally with two North American cultural regions -- the midwest [hi there Minnesota, we love you!] and (at least parts of) New England.
This isn't really because of TV or cultural export but because of real population origins and movements. We say "pop" and have "Canadian raising" in our speech, and so do people from Minnesota or Wisconsin and that doesn't come from TV or radio or movies. It comes from being neighbours and descended from the same population groups. The border also used to be a lot more porous. My mother's mother were (German descendant) North Dakotans who just basically popped across the border and started farming and lived in Sask and Alberta... without a lot of legal hassle at all.
Overtop of that, yes, there is a whole set of other cultural/legal/economic overlays, and media is a big part of that.
But this is still a regional story-- there's I think more in common culturally between e.g. regular families in Ontario and Wisconsin than there is between Wisconsin and Florida.
From that perspective, I have rarely fallen back to Canadian nationalism. I would in fact have been more in favour of a stronger union between some US states and the US economy and Canada -- in the past. But events of the last year have made clear what many our Loyalist ancestors already tried to warn us about 200 years ago: there is a dark and frankly kind of insane undercurrent in American political culture, and the foundations of the Canadian state are anything but artificial, they are based on an entirely different perspective on governance and culture because there's something kinda messed up in the kernel of the American conception of governance.
Which I think the people of Minnesota are seeing right now.
What was arrogant about my post? My point was that Canadians (like you) often exhibit this baffling, unfounded sense of superiority, one completely at odds with cold, hard migration data, as well as the habits of successful Canadians. It's especially baffling in light of Canada's "lost decade" of flat real GDP growth. Canada now has a per capita GDP that's lower than all but one or two U.S. states, and its economic "growth" is just people selling overpriced houses back and forth to each other. Yet the arrogance and condescending attitude towards their southern neighbor remains.
The big difference is the falkland islands are populated by brits loyal to britain whereas Greenland is populated by greenlanders who hate denmark because the danes committed many acts of genocide against greenlanders.
> I think most Americans are much more mobile and not used to the idea that someone could be strongly attached to an area as their home.
You think the people in the falklands are "native" to the falkland islands?
There are people who have been born and brought up in the Falkland Islands and have connections going back generations. They have been there longer than anyone else, and the islands had no previous indigenous population before France first colonised them.
As to whether they are native, that is a whole other can of worms, but they are more rooted to there than people who live in continental South America. Geographically they can't claim to be British but by sympathy they are. Things were shifting in Argentina's favour until the invasion.
By the way, this does apply to a certain percentage of Greenlanders. There are a few European Greenlanders, or people of recent mixed heritage so we could make similar arguments about them.
> polls also consistently show that Greenlanders do not want independence if the price is the collapse of the Greenlandic welfare state.
Even replacing independence with becoming part of the US - any one time payment couldn't ensure the continuation of the welfare state. That is, free healthcare, free higher education, access to social support, etc.
> > Even replacing independence with becoming part of the US - any one time payment couldn't ensure the continuation of the welfare state. That is, free healthcare, free higher education, access to social support, etc.
Yes if the one time payment is then invested in a big fund like Norway has, or considering the outcome that is at play here...more like California has with its CalPers
Of course it means that the actual money would be invested in US companies , hence subject to expropriation of the stock
And even the actual money if in dollars they can be taken away at any time
Wyoming or Vermont at 1 million each would be sub trillion range. Seems like reasonable. Technically they could just transfer all the debt they hold to residents there and get deal done.
I'm too lazy, but I hope someone is less lazy and would set up a Kickstarter and collect funds to be given to every resident of US Virgin Islands once they rejoined[1] Denmark. Would happily chip in a hundred or two for a good cause.
[1] Just learned that US Virgin islands are ex Danish colony US bought from Denmark about a century ago. In that agreement United States recognized Denmark's control over Greenland. Funny that.
They would have declined, as it's obviously a bad deal. European TV channels are interviewing Greenlanders these days asking the same question. They don't want it.
Furthermore, what you are suggesting is literally a mafia practice - sell us your business/property for an unfair price or we'll take it by force anyway.
I’m not sure why anyone is surprised that trump is acting like a mafia boss trying to shake down the rest of the world. This is who he has always been, the first time around there were just more people to say no to him.
Greendland People's Republic? Actually they were American all along, USA needs to save them from nazi Denmark. Grab it quick, before Russia does its usual scheme.
Even if every person on Greenland wanted to join the US, could they? Isn't the Land property of every Dane, not just the people currently living there?
If Greeland declares independence, Danish army would arrest all the local resisting people for terrorism, unlawful orders, etc (or shoot at them if needed).
They don't have a choice, they can't split away from Denmark right now, they are like in jail.
If someone tells you that Greenland can declare independence tomorrow without consequences from Denmark this is not true. They would first need to do great PR "we want freedom" and spread in the news that Denmark is evil and forgot about them, etc.
In 1979, Greenland achieved Home Rule, which included the formation of the Greenlandic Parliament, and it gained self-rule in 2009 through the passage of a law that included a ‘blueprint’ for seeking independence. The 2009 law firmly established that the decision to go for independence from Denmark would now rest with the Greenlandic people.
There is no doubt that the majority of Greenlanders want to use this option eventually. Polls show this. Independence has been accepted in Denmark as well. However, polls also consistently show that Greenlanders do not want independence if the price is the collapse of the Greenlandic welfare state.
Of course, you can't do that, there are criminal consequence: yes you can get arrested for that. Like in any country.
There are even worse: financial consequences (600M USD lost per year!).
But what if a richer buddy offers you protection and more money ?
In that specific agreement:
> "The agreement on independence shall be endorsed by a referendum in Greenland. The agreement shall furthermore be subject to the consent of the Folketing [Danish Parliament]."
It's "yes, you can leave, but you need our permission".
Today, Danish parliament is not really happy at the idea of giving away Greenland to anyone.
1) They want a new source of funding, ideally one they develop on their own (e.g. a mining and refining industry), to maintain their welfare state. It's a preference of theirs, not something imposed on them by the evil Danes.
Finding a new "buddy" to replace Denmark makes no sense. Why would they want to swap their dependence on a country which likes its welfare state (and is demonstrably good at administering it) for one which takes a notoriously dim view on such things?
2) Greenland becoming independent implies changing the borders of the kingdom of Denmark. That obviously requires a decision by parliament, no way around it.
Anyone interested in the facts can see the law in question here:
It would obviously not exist if Denmark was hellbent on denying Greenland its independence. All it does is lay out an orderly and straightforward process for the transition.
Factually, I agree with you on both points. Like from the rule of law perspective.
Statistically you are right, but in practice I would be cautious. I'm betting on the fact that a mad world is going to be even more mad (I couldn't imagine US threatening to invade Denmark... though I understand the US opportunity as well).
it would be a million times more probable that they would throw money to their friends in the military complex to take things by force, than to give money to strangers.
I don't have kids, but if I did, I would probably not go for Lego but for 3D-Printing.
It shouldn't be hard to print pieces that can snap together. In all kinds of colors and all kinds of forms.
I imagine the adventure of printing new pieces would be a fun thing for the kids and the parents. And when the kids are old enough, they can print pieces on their own. And a bit later design pieces on their own.
This. Even resin printers only get you kind of in the neighborhood of tolerance. Lego primarily uses injection molding for their parts. Their molds are insanely tight and low tolerance. One of the key costs of Lego bricks is the lifecycle of the molds. They don't last forever and lose tolerance over the course of several hundred thousand injections. Managing these molds and the sheer variety of parts they produce borders on logistical insanity. It is one of the most impressive logistics operations on the planet. I can build a functional car with fewer discrete pieces than large modern lego sets.
Not only tolerance, but also the fact that fused deposition is just not as accurate/dense/strong as injection molding when it comes to the building-blocks application.
disclaimer: i'm not a materials scientist, just a tinkerer who 3D prints and wishes they had the capability to do injection molding.
You can easily print bricks that work. They will just require more force to assemble than normal because you have to make them slightly undersized to make up for the lower tolerance.
Just think of how many 3d prints you've seen that consist of multiple parts friction for together.
> Just think of how many 3d prints you've seen that consist of multiple parts friction for together.
I've seen probably 10s, ranging from amateur-who-just-unpacked-their-printer to acquaintance who runs a business doing 3D printed products, and none of them come close to the experience of lego bricks, so far I'm not sure I'd actually call it "work". Stack 10 of these "custom" lego bricks and place them next to another stack of 10, and they almost certainly won't be as aligned as proper lego bricks, not to mention the whole thing will fall apart a lot easier.
Also: try taking them apart a hundred times and sticking them together again. If the parts initially stuck together strongly, chances are one of the parts will break down.
Printed pieces typically only have a single purpose, and lego-like snap together tolerances are hard to get working right with a 3d printer. Hell, they're hard to get right for the mass produced lego no-name duplicates. It can be fun to design shapes in tinkercad, but not as accessible for small children as just putting plastic bricks together.
I actually think what you suggest _would_ be brilliant if there was a printer that printed as nice and detailed parts as Lego does from ABS. The digital ecosystem for that would be crazy.
But.
Modern consumer printers are way better than decade ago but they still sort of suck if you want any fine details.
"It shouldn't be hard to print pieces that can snap together."
It's actually quite hard to print pieces that are functional and look nice.
Modern consumer 3D printers sort of suck for small details still. If all you print are Lego Dublo sized parts. And print them from ABS. You might succeed _sometimes_.
PLA the cheapest default plastic for filaments for extruders loses fit quite fast (I've tried). So ball joints etc will get loose pretty soon.
"Would there be any downside to this approach?"
Well, the adventure currently is the printing part and it's mostly not fun but one of those activities masochistic engineers (like myself) take up as a hobby.
The consumer 3D printers are improving! Maybe one day. But the material physics are not that comforting there.
Can confirm everything. PLA is completely unusable for this as it quickly deforms under constant pressure, so it is impossible to have a stable press fit with it. ABS would be the obvious choice (since Lego is ABS), but it's difficult to print. Generally, a press fit with ABS that can be handled by kids (so easy enough to create and remove), but still being sufficiently stable so that it can be handled, requires extremely tight tolerances which you will not be able to achieve with an FDM printer. Even very good FDM printers with small nozzles will have dozens of micrometers in tolerance, which is too large - pieces will either be almost impossible to fit, or they will just fall apart at the slightest movement. Resin printing is better, but again, the material is too soft and will not be able to withstand the pressures long-term. Even if you use special durable resin, it will deform quite quickly under constant pressure.
Not to discourage you, but it sounds like you'd be getting into a nerdy programmer dad hobby instead of just giving your kids toys. I doubt your children would be interested in watching you for hundreds of hours while you learn to use 3d modeling software and debug printer feed speeds. And once they were old enough (10-13+) to appreciate technical slogs, why wouldn't they do something cooler like make actual robots instead of reproducing a toy that you can buy a much better version of for $10?
3D printing would be good at making figurines and such, but you can't easily replicate the Lego system's modularity without their high tolerances.
That being said, it should be feasible to make something that allows easily programming Arduino and raspberry pi to interact with legos, similar to how their Mindstorms line worked. That would be the best of both worlds.
There are off-brand budget Lego blocks available and they're (in my experience) all awful. Legos are very precisely manufactured to fit together smoothly, if the off-brand ones made in a factory can't replicate that then I don't have much hope for small-scale 3D printing.
The gap is reducing significantly - I have collected and built Lego for a long time and had this opinion but have recently discovered Lumibricks (formerly Funwhole) - excellent designs, and around 1/2 the price of Lego (but they all include lighting elements) and having put them together I can say they feel exactly like Lego. I believe there are other brands of similar quality.
No sure what exactly you mean by "off-brand", but I have recently bought big sets from Mould King and Cada and they are perfectly fine. Not only are they cheap, I'd say the color consistency is even a bit better. The sets themselves are great, really creative, challenging to build even for older kids, something Lego stopped doing many years ago.
This is not necessarily true. It is possible for all real numbers (and indeed all mathematical objects) to be definable under ZFC. It is also possible for that not to be the case. ZFC is mum on the issue.
Basically you can't do a standard countability argument because you can't enumerate definable objects because you can't uniformly define "definability." The naive definition falls prey to Liar's Paradox type problems.
I think you're overthinking it. Define a "number definition system" to be any (maybe partial) mapping from finite-length strings on a finite alphabet to numbers. The string that maps to a number is the number's definition in the system. Then for any number definition system, almost all real numbers have no definition.
Sure, you can do that. The parent's point is that if you want this mapping to obey the rules that an actual definition in (say) first-order logic must obey, you run into trouble. In order to talk about definability without running into paradoxes, you need to do it "outside" your actual theory. And then statements about cardinalities - for example "There's more real numbers than there are definitions." - don't mean exactly what you'd intuitively expect. See the result about ZFC having countable models (as seen from the "outside") despite being able to prove uncountable sets exist (as seen from the "inside").
No, this is a standard fallacy that is covered in most introductory mathematical logic courses (under Tarski's undefinability of truth result).
> Define a "number definition system" to be any (maybe partial) mapping from finite-length strings on a finite alphabet to numbers.
At this level of generality with no restrictions on "mapping", you can define a mapping from finite-length strings to all real numbers.
In particular there is the Lowenheim-Skolem theorem, one of its corollaries being that if you have access to powerful enough maps, the real numbers become countable (the Lowenheim-Skolem theorem in particular says that there is a countable model of all the sets of ZFC and more generally that if there is a single infinite model of a first-order theory, then there are models for every cardinality for that theory).
Normally you don't have to be careful about defining maps in an introductory analysis course because it's usually difficult to accidentally create maps that are beyond the ability of ZFC to define. However, you have to be careful in your definition of maps when dealing with things that have the possibility of being self-referential because that can easily cross that barrier.
Here's an easy example showing why "definable real number" is not well-defined (or more directly that its complement "non-definable real number" is not well-defined). By the axiom of choice in ZFC we know that there is a well-ordering of the real numbers. Fix this well-ordering. The set of all undefinable real numbers is a subset of the real numbers and therefore well-ordered. Take its least element. We have uniquely identified a "non-definable" real number. (Variations of this technique can be used to uniquely identify ever larger swathes of "non-definable" real numbers and you don't need choice for it, it's just more involved to explain without choice and besides if you don't have choice, cardinality gets weird).
Again, as soon as you start talking about concepts that have the potential to be self-referential such as "definability," you have to be very careful about what kinds of arguments you're making, especially with regards to cardinality.
Cardinality is a "relative" concept. The common intuition (arising from the property that set cardinality forms a total ordering under ZFC) is that all sets have an intrinsic "size" and cardinality is that "size." But this intuition occasionally falls apart, especially when we start playing with the ability to "inject" more maps into our mathematical system.
Another way to think about cardinality is as a generalization of computability that measures how "scrambled" a set is.
We can think of indexing by the natural numbers as "unscrambling" a set back to the natural numbers.
We begin with complexity theory where we have different computable ways of "unscrambling" a set back to the natural numbers that take more and more time.
Then we go to computability theory where we end up at non-computably enumerable sets, that is sets that are so scrambled that there is no way to unscramble them back to the natural numbers via a Turing Machine. But we can still theoretically unscramble them back to the natural numbers if we drop the computability requirement. At this point we're at definability in our chosen mathematical theory and therefore cardinality: we can define some function that lets us do the unscrambling even if the actual unscrambling is not computable. But there are some sets that are so scrambled that even definability in our theory is not strong enough to unscramble them. This doesn't necessarily mean that they're actually any "bigger" than the natural numbers! Just that they're so scrambled we don't know how to map them back to the natural numbers within our current theory.
This intuition lets us nicely resolve why there aren't "more" rational numbers than natural numbers but there are "more" real numbers than natural numbers. In either case it's not that there's "more" or "less", it's just that the rational numbers are less scrambled than the real numbers, where the former is orderly enough that we can unscramble it back to the natural numbers with a highly inefficient, but nonetheless computable, process. The latter is so scrambled that we have no way in ZFC to unscramble them back (but if you gave us access to even more powerful maps then we could scramble the real numbers back to the natural numbers, hence Lowenheim-Skolem).
It doesn't mean that in some deep Platonic sense this map doesn't exist. Maybe it does! Our theory might just be too weak to be able to recognize the map. Indeed, there are logicians who believe that in some deep sense, all sets are countable! It's just the limitations of theories that prevent us from seeing this. (See for example the sketch laid out here: https://plato.stanford.edu/entries/paradox-skolem/#3.2). Note that this is a philosophical belief and not a theorem (since we are moving away from formal definitions of "countability" and more towards philosophical notions of "what is 'countability' really?"). But it does serve to show how it might be philosophically plausible for all real numbers, and indeed all mathematical objects, to be definable.
I'll repeat Hamkins' lines from the Math Overflow post because they nicely summarize the situation.
> In these pointwise definable models, every object is uniquely specified as the unique object satisfying a certain property. Although this is true, the models also believe that the reals are uncountable and so on, since they satisfy ZFC and this theory proves that. The models are simply not able to assemble the definability function that maps each definition to the object it defines.
> And therefore neither are you able to do this in general. The claims made in both in your question and the Wikipedia page [no longer on the Wikipedia page] on the existence of non-definable numbers and objects, are simply unwarranted. For all you know, our set-theoretic universe is pointwise definable, and every object is uniquely specified by a property.
I think I understand your argument (you could define "the smallest 'undefinable' number" and now it has a definition) but I still don't see how it overcomes the fact that there are a countable number of strings and an uncountable number of reals. Can you exhibit a bijection between finite-length strings and the real numbers? It seems like any purported such function could be diagonalized.
My other reply is so long that HN collapsed it, but addresses your particular question about how to create the mapping between finite-length strings and the real numbers.
Here's another lens that doesn't answer that question, but offers another intuition of why "the fact that there are a countable number of strings and an uncountable number of reals" doesn't help.
For convenience I'm going to distinguish between "collections" which are informal groups of elements and "sets" which are formal mathematical objects in some kind of formal foundational set theory (which we'll assume for simplicity is ZFC, but we could use others).
My argument demonstrates that the "definable real numbers" is not a definition of a set. A corollary of this is that the subcollection of finite strings that form the definitions of unique real numbers is not necessarily an actual subset of the finite strings.
Your appeal that such definitions are themselves clearly finite strings is only enough to demonstrate that they are a subcollection, not a subset. You can only demonstrate that they are a subset if you could demonstrate that the definable real numbers form a subset of the real numbers which as I prove you cannot.
Then any cardinality arguments fail, because cardinality only applies to sets, not collections (which ZFC can't even talk about).
After all, strictly speaking, an uncountable set does not mean that such a set is necessarily "larger" than a countable set. All it means is that our formal system prevents us from counting its members.
There are subcollections of the set of finite strings that cannot be counted by any Turing Machine (non-computably enumerable sets). It's not so crazy that there might be subcollections of the set of finite strings that cannot be counted by ZFC. And then there's no way of comparing the cardinality of such a subcollection with the reals.
Another way of putting it is this: you can diagonalize your way out of any purported injection between the reals and the natural numbers. I can just the same diagonalize my way out of any purported injection between the collection of definable real numbers and the natural numbers. Give me such an enumeration of the definable real numbers. I change every digit diagonally. This uniquely defines a new real number not in your enumeration.
Perhaps even more shockingly, I can diagonalize my way out of any purported injection from the collection of finite strings uniquely identifying real numbers to the set of all natural numbers. You purport to give me such an enumeration. I add a new string that says "create the real number such that the nth digit is different from the real number of the nth definition string." Hence such a collection is an uncountable subcollection of a countable set.
> Can you exhibit a bijection between finite-length strings and the real numbers? It seems like any purported such function could be diagonalized.
Let's start with a mirror statement. Can you exhibit an bijection between definitions and the subset of the real numbers they are supposed to refer to? It seems like any purported such bijection could be made incoherent by a similar minimization argument.
In particular, no such function from the finite strings to the real numbers, according to the axioms of ZFC can exist, but a more abstract mapping might. In much the same way that no such function from definitions to (even a subset of) the real numbers according to the axioms of ZFC can exist, but you seem to believe a more abstract mapping might.
I think your thoughts are maybe something along these lines:
"Okay so fine maybe the function that surjectively maps definitions to the definable real numbers cannot exist, formally. It's a clever little trick that whenever you try to build such a function you can prove a contradiction using a version of the Liar's Paradox [minimality]. Clearly it definitely exists though right? After all the set of all finite strings is clearly smaller than the real numbers and it's gotta be one of the maps from finite strings to the real numbers, even if the function can't formally exist. That's just a weird limitation of formal mathematics and doesn't matter for the 'real world'."
But I can derive an almost exactly analogous thing for cardinality.
"Okay so fine maybe the function that surjectively maps the natural numbers to the real numbers cannot exist, formally. It's a clever little trick that whenever you try to build such a function you can prove a contradiction using a version of the Liar's Paradox [diagonalization]. Clearly it definitely exists though right? After all the set of all natural numbers is clearly just as inexhaustible as the real numbers and it's gotta be one of the maps from the natural numbers to the real numbers, even if the function can't formally exist. That's just a weird limitation of formal mathematics and doesn't matter for the 'real world'."
I suspect that you feel more comfortable with the concept of cardinality than definability and therefore feel that "the set of all finite strings is clearly 'smaller' than the real numbers" is a more "solid" base. But actually, as hopefully my phrasing above suggests, the two scenarios are quite similar to each other. The formalities that prevent you from building a definability function are no less artificial than the formalities that prevent you from building a surjection from the natural numbers to the real numbers (and indeed fundamentally are the same: the Liar's Paradox).
So, to understand how I would build a map that maps the set of finite strings to the real numbers, when no such map can formally exist in ZFC, let's begin by understanding how I would rigorously build a map that maps all sets to themselves (i.e. the identity mapping), even when no such map can formally exist as a function in ZFC (because there is no set of all sets).
(I'm choosing the word "map" here intentionally; I'll treat "function" as a formal object which ZFC can prove exists and "map" as some more abstract thing that ZFC may believe cannot exist).
We'll need a detour through model theory, where I'll use monoids as an illustrative example.
The definition of an (algebraic) monoid can be thought of as a list of logical axioms and vice versa. Anything that satisfies a list of axioms is called a model of those axioms. So e.g. every monoid is a model of "monoid theory," i.e. the axiomos of a monoid. Interestingly, elements of a monoid can themselves be groups! For example, let's take the set {{}, {0}, {0, 1}, {0, 1, 2}, ...}, as the underlying set of a monoid whose monoid operation is just set union and whose elements are all monoids that are just modular addition.
In this case not only is the parent monoid a model of monoid theory, each of its elements are also models of monoid theory. We can then in theory use the parent monoid to potentially "analyze" each of its individual elements to find out attributes of each of those elements. In practice this is basically impossible with monoid theory, because you can't say many interesting things with the monoid axioms. Let's turn instead to set theory.
What does this mean for ZFC? Well ZFC is a list of axioms, that means it can also be viewed as a definition of a mathematical object, in this case a set universe (not just a single set!). And just like how a monoid can contain elements which themselves are monoids, a set universe can contain sets that are themselves set universes.
In particular, for a given set universe of ZFC, we know that in fact there must be a countable set in that set universe, which itself satisfies ZFC axioms and is therefore a set universe in and of itself (and moreover such a countable set's members are themselves all countable sets)!
Using these "miniature" models of ZFC lets us understand a lot of things that we cannot talk about directly within ZFC. For example we can't make functions that map from all sets to all sets in ZFC formally (because the domain and the codomain of a function must both be sets and there is no set of all sets), but we can talk about functions from all sets to all sets in our small countable set S which models ZFC, which then we can use to potentially deduce facts about our larger background model. Crucially though, that function from all sets to all sets in S cannot itself be a member of S, otherwise we would be violating the axioms of ZFC and S would no longer be a model of ZFC! More broadly, there are many sets in S, which we know because of functions in our background model but not in S, must be countable from the perspective of our background model, but which are not countable within S because S lacks the function to realize the bijection.
This is what we mean when we talk about an "external" view that uses objects outside of our miniature model to analyze its internal objects, and an "internal" view that only uses objects inside of our miniature model.
Indeed this is how I can rigorously reason about an identity map that maps all sets to themselves, even when no such identity function exists in ZFC (because again the domain and codomain of a function must be sets and there is no set of all sets!). I create an "external" identity map that is only a function in my external model of ZFC, but does not exist at all in my set S (and hence S can generate no contradiction to the ZFC axioms it claims to model because it has no such function internally).
And that is how we can talk about the properties of a definability map rigorously without being able to construct one formally. I can construct a map, which is a function in my external model but not in S, that maps the finite strings of S (encoded as sets, as all things are if you take ZFC as your foundation) that form definitions to some subset of the real numbers in S. But there's multiple such maps! Some maps that map the finite strings of S to the real numbers "run out of finite strings," but we know that all the elements of S are themselves countable, which includes the real numbers (or at least S's conception of the real numbers)! Therefore, we can construct a bijective mapping of the finite strings of S to the real numbers of S. Remember, no such function exists in S, but this is a function in our external model of ZFC.
Since this mapping is not a function within S, there is no contradiction of Cantor's Theorem. But it does mean that such a mapping from the finite strings of S to the real numbers of S exists, even if it's not as a formal function within S. And hence we have to grapple with the problem of whether such a mapping likewise exists in our background model (i.e. "reality"), even if we cannot formally construct such a mapping as a function within our background model.
And this is what I mean when I say it is possible for all objects to have definitions and to have a mapping from finite strings to all real numbers, even no such formal function exists. Cardinality of sets is not an absolute property of sets, it is relative to what kinds of functions you can construct. Viewed through this lens, the fact that there is no satisfiability function that maps definitions to the real numbers is just as real a fact as the fact that there is no surjective function from the natural numbers ot the real numbers. It is strange to say that the former is just a "formality" and the latter is "real."
For more details on all this, read about Skolem's Paradox.
Leads to really fun statements like "there exists a proof that all reals are equal to themselves" and "there does not exist a proof for every real number that it is equal to itself" (because `x=x`, for most real numbers, can't even be written down, there are more numbers than proofs).
Any HN comment is a finite expression, so it's impossible for me to specify a particular one. But the number of finite expressions is countable, and the number of reals is vastly more than a countable number, so most reals cannot be described in any human sense.
I think (I am not a mathematician) that depends on whether you accept non-constructive proofs as valid. Normally you reason that any mapping from natural numbers onto the reals is incomplete (eg Cantor's argument), and that the sets of computable or describable numbers are countable, and therefore there exist indescribable real numbers. But if you don't like that last step, you do have company:
There are more infinite sequences than finite ones.
So not all infinite sequences can be uniquely specified by a finite description.
Like √2 is a finite description, so is the definition of π, but since there is no way to map the abstract set of "finite description" surjectively to the set of infinite sequences you find that any one approach will leave holes.
But doesn't this assume what you intend to show? Of course you can't specify an infinite and non-repeating sequence, but how do you know that is a number?
Slightly longer answer decimal numbers between 0 and 1 can be written as the sum of a_0*10^0 + a_1*10^1 + a_2*10^2 + ... + a_i*10^i + ... where a_i is one of 0,1,2,3,4,5,6,7,8,9. for series in this shape you can prove that the sum of two series is the same iff and only if the sequence of digits are all the same (up to the slight complication of 0.09999999 = 0.1 and similar)
You can't know. However, it is a consequence of the axiom of choice (AC). You can't know if AC is true either; but mathematics without it is really really hard, so it usually assumed.
You can't actually pick real numbers at random. You especially can't do it on a computer, since all numbers representable in a finite number of digits or bits are rational.
It's true that no matter what symbolic representation format you choose (binary or otherwise) it will never be able to encode all irrational numbers, because there are uncountably many of them.
But it's certainly false that computers can only represent rational numbers. Sure, there are certain conventional formats that can only represent rational numbers (e.g. IEEE-754 floating point) but it's easy to come up with other formats that can represent irrationals as well. For instance, the Unicode string "√5" is representable as 4 UTF-8 bytes and unambiguously denotes a particular irrational.
> the Unicode string "√5" is representable as 4 UTF-8 bytes
As the other person pointed out, this is representing an irrational number unambiguously in a finite number of bits (8 bits in a byte). I fail to see how your original statement was careful :)
> representable in a finite number of digits or bits
Isn't "unambiguous representation" impossible in practice anyway ? Any representation is relative to a formal system.
I can define sqrt(5) in a hard-coded table on a maths program using a few bytes, as well as all the rules for manipulating it in order to end up with correct results.
Well yeah but if we’re being pedantic anyway then “render these bits in UTF-8 in a standard font and ask a human what number it makes them think of” is about as far from an unambiguous numerical representation as you could get.
Of course if you know that you want the square root of five a priori then you can store it in zero bits in the representation where everything represents the square root of five. Bits in memory always represent a choice from some fixed set of possibilities and are meaningless on their own. The only thing that’s unrepresentable is a choice from infinitely many possibilities, for obvious reasons, though of course the bounds of the physical universe will get you much sooner.
Exactly right. You can pick and use real numbers, as long as they are only queried to finite precision. There are lots of super cool algorithms for doing this!
Kind of, but you're not just picking rationals, you're picking rationals that are known to converge to a real number with some continuous property.
You might be interested in this paper [1] which builds on top of this approach to simulate arbitrarily precise samples from the continuous normal distribution.
Not really. You can simulate a probability of 1/x by expanding 1/x in binary and flipping a coin repeatedly, once for each digit, until the coin matches the digit (assign heads and tails to 0 and 1 consistently). If the match happened on 1, then it's a positive result, otherwise negative. This only requires arbitrary but finite precision but the probability is exactly equal to 1/x which isn't rational.
You seem to be positing that Maxwell's Demon can be reassigned to another impossible task, but that isn't a proper use of his "powers".
Infinities defy simple assumptions about maths, while Maxwell's Demon only needs to ignore the Laws of Thermodynamics.
I'm being serious, not glib, here. "And then do it infinitely many times" doesn't automatically enable any possible outcome, any more than the "multiverse of all possible outcomes" enables hot dog fingers on Jamie Curtis.
Not really. In all of our physical theories, curved paths are actual curves. So, (assuming circular orbits for a second) the ratio between the length of the Earth's orbit around the Sun and the distance between the Earth and the Sun is Pi - so, either the length of the path or the straight line distance must be an irrational number. While the actual orbit is elliptical instead of circular, the relation still holds.
Of course, we can only measure any quantity up to a finite precision. But the fact that we chose to express the measurement outcome as 3.14159 +- 0.00001 instead of expressing it as Pi +- 0.00001 is an arbitrary choice. If the theory predicts that some path has length equal exactly to 2.54, we are in the same situation - we can't confirm with infinite precision that the measurement is exactly 2.54, we'll still get something like 2.54 +- 0.00001, so it could very well be some irrational number in actual reality.
and, depending on how you define the rationals, -0.
https://en.wikipedia.org/wiki/Integer: “An integer is the number zero (0), a positive natural number (1, 2, 3, ...), or the negation of a positive natural number (−1, −2, −3, ...)”
According to that definition, -0 isn’t an integer.
Combining that with https://en.wikipedia.org/wiki/Rational_number: “a rational number is a number that can be expressed as the quotient or fraction p/q of two integers, a numerator p and a non-zero denominator q”
means there’s no way to write -0 as the quotient or fraction p/q of two integers, a numerator p and a non-zero denominator q.
This is how old temperature-noise based TRNGs can be attacked (modern ones use a different technique, usually a ring-oscillater with whitening... although i have heard noise-based is coming back but i've been out of the loop for a while)
Well, sampling is technically an analog operation that is separate from the conversion operation that makes the result digital. But then analog circuits don't ever actually hold a single real number, in practice there is always noise and that in practice limits the precision to less than what can be fairly easily achieved digitally.
Sure, but we are talking about generating a random number, not sampling noise: those are two different things, albeit the former can be derived from the latter but not directly and as simply as the parent post claimed. Just sampling analog noise does not generate a "true" random number that can satisfy a set of design parameters to configure the NIST 800-90b entropy assessment (well, one could pick shitty parameters for the probability tests, but let's assume experts at the helm). Hence the need for software whitening.
(^^^ this is a fun tool, I recommend playing with it to learn how challenging it is to generate "true" random numbers.)
An infinite precision ADC couldn't be subject to thermal attack because you could just sample more bits of precision. (Of course, then we'd be down to Planck level precision so obviously there are limits, but my point still stands, at least _I_ think it does. :))
Use an analog computer how, to do what? An analog computer can do analog operations on analog signals, but you can't get an irrational number out of it ... this can be viewed as a sort of monad.
It says the input has 3 dimensions, two spatial dimensions and one feature dimension. So it would be a 2D grid of numbers. Like a grayscale photo. But at 00:38 it shows the numbers and it looks like each of the blocks positioned in 3D space holds a floating-point value. Which would make it a 4-dimensional input.
I have yet to see a good web based text editor with syntax highlighting. They all mess with the native search functionality of the browser. Because they can't just use a textarea for the edit area. With this approach, it would be possible.
I wonder how usable a Python version of this would be?
I have yet to see a good web based text editor with syntax highlighting.
I slightly expect you to pull a "no true Scotsman" here and suggest it's actually no good because it doesn't really support mobile browsers very well, but Microsoft's Monaco editor that's driven from VS Code is quite good. https://microsoft.github.io/monaco-editor/
It seems to have the same problems all of the web based editors I have seen have. Either they capture ctrl+f and take away the native search experience. Or they have a broken search experience. This one is in the latter category.
When I hit ctrl+f on that page and type "export":
First it says "1 of 4 matches" but nothing is highlighted.
When I hit enter, it says "2 of 4 matches" and again, nothing is highlighted.
When I hit enter again, it says "3 of 4 matches" and the first match is highlighted.
When I hit enter again, it says "4 of 4 matches" and the second match is highlighted.
Contenteditable plus the CSS Custom Highlight API (which highlights ranges instead of elements) might indeed allow for a good solution. But I have not yet seen an editor that does that.
But does it support editing highlighted text? If not, one would have to do some trickery by hiding a textarea and updating the <code> element on each keypress, I guess. Which probably has a thousand corner cases one would have to deal with.
The example on their website is editable and it looks like they overlay the highlighted output on top of the textarea with `pointer-events: none` like you mentioned.
Hmm .. and the approach already shows its weaknesses when I play with it: When I search for something on the page, it gives me twice as many hits as there are. And jumps around two times to each hit when I use the "next" button.
There is a neat `inert` html attribute you can use to disable all interactions as well as hide the text from ctrl+f searches. (Sadly Safari is the weird one out, and does not exclude the content from searches.)
I just wanted to write about a similar observation when using it in FireFox on Linux:
When wiggle the spring, keep the mouse inside the white area until it is at rest, press CTRL+u to see the source code, move the mouse to close the source code tab and close it - for some magical reason the spring is moving again for a little bit.
HOW TO RETURN words document:
PUT {} IN collection
FOR line IN document:
FOR word IN split line:
IF word not.in collection:
INSERT word IN collection
RETURN collection
In Python it would be:
def words(document):
collection = set()
for line in document:
for word in line.split():
if word not in collection:
collection.add(word)
return collection
I kept the splitting by line and "if word not in collection:" in there even though they don't have an impact on the outcome. I have the feeling that even in the original example they have only been put there to show the language constructs, not to do anything useful. If one wanted to optimize it, it could all be collapsed to just "return set(text.split())", but that would not show off the language features.
ABC uses 225 chars, Python 218 chars. 3% less.
So one could say Python is 3% more efficient than ABC.
As I understand it, "RETURN" means that the function will return something. And that when you define a function that returns nothing, but only does something, you just use "HOW TO".
> public static function words(string $document): array {
>, though is verbose, at least means something
I disagree, it is hardly verbose. there can be further detailing of the input and return value of the function beyond a simple type. Java didn't go far enough, not by a long shot - we need hardcore painstaking, completely brutal typing. Such typing will be highly beneficial for software reliability - you're not against reliability are you?
That seems like not much compared to the hundreds of billions of dollars US companies currently invest into their AI stack? OpenAI pays thousands of engineers and researchers full time.
It is. The problem is latency. All these fields are moving very fast, and so it doesn't sound bad spending 6 months tuning something, but in reality what is happening is that during those 6 months the guy who built the thing you're tuning has iterated 5 more times and what you started on 6 months ago is now much much better than what you got handed 6 months ago whilst simultaneously being much worse than what that person has in their hands today. If the field you're working in is relatively static, or your performance gap is large enough it makes sense. But in most fields the performance gap is large in absolutely terms but small in temporal terms. You could make something run 10x faster, but you can't build something that will run faster than what will be state of the art in 2 months.
That would be just $50B - a pretty cheap way to increase the size of the USA by 20%.
reply