So he set out to compare the natural numbers and the real numbers between zero and one. Cantor started by assuming he could perfectly map these sets to each other, one-to-one. So he imagined writing down an infinite list with a natural number on one side and a real number between zero and one on the other.
to infinity. Cantor had revealed something remarkable. Infinity doesn't come in just one size.
These infinities like the set of all real numbers, the complex numbers, they can't be matched one-to-one with the natural numbers. Cantor's results rocked the mathematical community. After all, how can something that continues forever be bigger than something else that continues forever?
There's a clear starting point, zero, and all their subsets also have a definitive starting point. Cantor had successfully well ordered a set that was infinite in both directions, but it was only countably infinite. In his next book, he published his well ordering theorem.
but there was one big reason that Cantor was so confident in his theorem. Cantor was a devout Lutheran and he believed God was speaking through him. He said, my theory stands as firm as a rock.
And Kronecker used to be Cantor's teacher. Cantor dreamed of joining him at the University of Berlin, but all his applications were mysteriously denied. So Cantor took the rejection personally.
So Cantor's well-ordering theorem and Zermelo's axiom of choice are equivalent. Cantor was so relieved. Zermelo had proved the well-ordering theorem and well ordered the real numbers all in under a month.
response parameter? And what's challenging about this is that the settings in which response parameters to wealth tax have been estimated, you know, they're things like Swiss cantons that have moved around the wealth tax rates. You're looking at to what extent, you know, wealthy individuals are sensitive to, you know, Swiss cantons Cantonal wealth taxes, which is a well-identified
argued against this sort of potentialist orthodoxy in The Dialogue of Two New Sciences. Really lovely account there that he gave. And that the... In many ways, Galileo was anticipating Cantor's developments, except he couldn't quite push it all the way through and ended up throwing up his hands in confusion in a sense. I mean, the Galileo paradox is the idea or the observation that if you think about the natural numbers,
Salimos, íbamos para casa y como ya eran la una de la mañana, del lado derecho de la calle se escuchaba la música de las cantinas familiares. Y del lado izquierdo de la de la calle se escuchaba la música de estos locales nocturnos que comenzaban a funcionar. Entonces, yo recuerdo que mi padre paró en un semáforo y yo les dije a los 2 miren, yo no sé de qué lado de la calle voy
Yo, al día de hoy, me despierto y canto desde que despierto, literalmente, hasta las últimas horas de la noche. Canto todo el día en casa, quien mi familia, que está aquí conmigo, me escucha cantar. Hago mis actividades cantando.
I always say that I'm lucky. Cantina. Great. I want to open it.
And we wound up spending the next nine years running a company called Cantora. Cantora developed and signed a handful of bands, hundreds of concerts, and eventually raised a technology fund where we were helping startups and their founders navigate the music industry. And on the side, I started experimenting with building community.
So I suppose you know nothing about them. Cantor did a wonderful thing in the decade-- Georg Cantor, famous German mathematician-- did a wonderful thing in the decade from 1870 to 1880. He founded the theory of transfinite numbers, or just infinite, if you like-- past the finites.
Everybody else stopped counting somewhere at the finite stage. Cantor actually copied the usual way of counting, starting at one. But it's nicer to count starting at zero.
Everybody stopped, say, at 2,417, or some other finite number. Cantor carried on, imagining counting an infinite collection, getting to the end, and then you have a problem of what the next number is after all the finite whole numbers. And Cantor originally called it infinity, but it doesn't behave like the infinity that people know from calculus.
I wondered whether Cantor had had similar feelings. Cantor eventually-- well, he was institutionalized several times and died in a lunatic asylum in Halle, which later became part of East Germany in 1918.
He used to say, he's turned out nice again, he-he-he. Cantalini's Salerno Beach Italian Restaurant.
Calibrating your palate at a certain sodium level. Cantaloupe. Cantaloupe. So what you were getting on your tongue, using just your sense of taste, was sweet because that's the only basic taste that's in jelly beans.
they were at that level. And each time those two pieces were designed to change every time. If you listen to Death Star Cantina you'll see me going what I don't who I'm searching for the words because I have I've decided not to choose the words and so it's constantly fluid which means that it always stays alive and I noticed in the early stages of doing
And, you know, basically I'll give you an example, I went to a conference in Shanghai, of university officials from all over the world and a French philosopher named Monique Canto-Sperber who is the president of the Ecole Normale Superieure, which is one of the elite grandes ecole in France, she came over to Shanghai to give a long impassioned speech about how terrible the Shanghai rankings are and they used the wrong inputs and they used the wrong measures and they don't do justice to wonderful places like hers. And
his homosexuality and and as a result he was tried and he was condemned to uh Cantor. This is the the place where he went to meet Cantor. This is if you the lower panel you see this is if you're
He can only do one. He cant ' do trial calculations. You make it so that when you send the public component of the Diffie Hellman calculation, you don't send it right away.
My kitchen's so embarrassing. I cant believe it.
He took on the task of putting the real numbers in a definitive order, even if it killed him and it nearly did. Georg Cantor was a talented German mathematician who found himself at the center of a firestorm after publishing one of his very first papers at the age of 29.
You can literally count them, one, two, three and so on. So Cantor called these countable infinities, but then there are bigger infinities, Cantor called them uncountable. These infinities like the set of all real numbers, the complex numbers, they can't be matched one-to-one with the natural numbers.
His success only spurred him to pursue his even grander goal to show that even uncountably infinite sets could be placed in a definitive order. What Cantor called a well-order. For a set to be well ordered, he required two conditions.
Integers stretch off to infinity in both the positive and negative directions. Well Cantor realized he could just pick zero as the starting point and from there his ordering went one, negative one, two, negative two, ranking the integers by their absolute value, their distance from zero.
Cantor dreamed of joining him at the University of Berlin, but all his applications were mysteriously denied. So Cantor took the rejection personally. In 1884, he wrote 52 letters to a friend and every one of them bemoaned Kronecker.
In 1884, he wrote 52 letters to a friend and every one of them bemoaned Kronecker. Soon Cantor suffered what would be the first of many nervous breakdowns. He was confined to a sanitarium for recovery.
The only way he could prove everyone wrong was by well ordering the real numbers, but he couldn't find a starting point, literally. Once Cantor was released from the sanatorium, he stepped away from math, a broken man. And over the next 15 years he taught philosophy and rarely dabbled in his old pursuits.
meaning all sets can be well ordered no matter the infinity. So Cantor's well-ordering theorem and Zermelo's axiom of choice are equivalent. Cantor was so relieved.
Right, "King of Love Songs." After "Jade Solid Gold," I found there could be two versions — a Cantonese one. George Lam's "12 Minutes 10 Inches?” "12 Minutes 12 Inches?” Actually, "12 Minutes 10 Inches" — Chow Yun-fat did it once.
So Cantinflas was a huge part of my childhood and also moved on to stuff later on like "Paper Moon." And that was a big one, which I shouldn't
Haber cantado con Bebe, que también creo que es una de las artistas iberoamericanas más potentes y con decir absolutamente original y bello.
Después cantas para miles de personas, ¿viste?
¿Qué es lo que hago? Es cantar, todo el día voy cantando, esa canción la voy cantando. Y aquella primera letra que viene, después lo que empiezo a hacer con estas herramientas de vocabulario que te decía antes que tengo, puedo empezar,
Les cantonnent dans des situations économiques et sociales, évidemment défavorables.
Aunque cantes feo, pero tú haz algo.
of Canterbury we've just been hearing about.
a Cantonese style. All right. Well, let me get out of your way. Well, it's not going to be too fancy here.
Cantor carried on, imagining counting an infinite collection, getting to the end, and then you have a problem of what the next number is after all the finite whole numbers. And Cantor originally called it infinity, but it doesn't behave like the infinity that people know from calculus. So after a time he rubbed a little bit of the infinity symbol off and took the Greek little omega as his notion for the first infinite number counting
It was unsolved in a way. But Cantor said in this paper in the 1940s-- Godel. Godel said, yeah. Sorry.
Some cantons are Catholic.
Some cantons are Protestant.
Which cantons have the most cows?
most Cantillon on tap in the world.
Leading the charge was Leopold Kronecker, the head of mathematics at the University of Berlin. Kronecker completely dismissed Cantor's work, labeling him a scientific charlatan and a corrupter of the youth. And Kronecker used to be Cantor's teacher.
Axioms are simple statements we accept as true without proof. Zermelo realized Cantor's assumption needed to be formalized into something that holds up in a system of proof. A new axiom that said, making all of those choices was possible.
Keeping going doesn't guarantee the perfect outcome you envision, I love Cantopop. Growing up abroad, it was Cantopop, it was your songs -Thank you. -that helped me understand this culture.