Cantor had revealed something remarkable. Infinity doesn't come in just one size.Some infinities like the set of square numbers, integers or rational numbers can be paired perfectly with the natural numbers.
you could reassemble them into two balls each identical to the one we started with, and you could keep going until eventually you have an infinite number of balls, infinity all from one.This sounds absurd, but we can actually see how it works by building a graph.
- The following is a conversation with Joel David Hamkins, a mathematician and philosopher specializing in set theory, the foundation of mathematics and the nature of infinity . He is the number one highest rated user on MathOverflow, which I think is a legendary accomplishment. MathOverflow, by the way, is like StackOverflowbut for research mathematicians. He is also the author of several books, including Proof in the Art of Mathematics and Lectures on the
back to Ancient Greek times when Aristotle emphasized the potential aspect of infinity as opposed to the impossibility, according to him, of achieving an actual infinity . And Archimedes' method of exhaustion where he is trying to understand the area of a region by carving it into more and more triangles, say, andsort of exhausting the area and thereby understanding the total area in terms of the sum of the areas of the pieces that he put into it.
one kind of infinity , then it wouldn't be surprising at all, because if you take two infinite sets and put them together, then it's still infinite. And so if there were only one kind of infinity , then it shouldn't be surprising...that the union of two countable sets is countable. So there's another way to push this a bit harder, and that is when Hilbert's train
I used to say that continuity is an approximation to many. Infinity is an approximation to many.Now there is always a finite number, so to say.
"Civil War" takes your heroes and pits them against each other in a fight. "Infinity War--" we kill half your favorite heroes.
A lot of your pants. Infinity . Cool. I can see I have some audience questions.
uh the market runs out of buyers and uh one of the things that that's been talked about in this ongoing Infinity which is what their plan was and what they were hoping was going to occur all
all mathematicians treated infinity sort of you know threw long sticks. Okay, they never touched it. Okay, they uh infinity was something that you write under the limit sign. Okay, so you know and you don't think about it anymore anduh and uh but Canour started to study infinity and dissect infinity and in
- Because even once you know how to use this chart, you're still left wondering how it pulls off its strangest bit of witchcraft: trapping infinity inside a finite circle. Well, to understand that, we have to go back 100 years to the origins of this chart.
That's why this is called Cantor's Diagonalization Proof and it shows there must be more real numbers between zero and one than there are natural numbers extending out to infinity . Cantor had revealed something remarkable.
You break the stick, you put it down, and you will respawn exactly where that insert was located. Discrete infinity is the idea that you have limited modules but infinite possibilities, right?
mathematician now and then deal dealt with infinity but u uh at Gaus himself had admonition all mathematicians treated infinity sort of you know threw long sticks. Okay, they never touched it. Okay, they uh
- And that's showing, throwing no shade on their colleague who did these cool experiments. It can be plus infinity .
- And that's showing, throwing no shade on their colleague who did these cool experiments. It can be minus infinity .
What is one thing AI should never decide? There's like infinity things that AI should never decide. Um it should never make judgments that can have a negative impact on people, I think. So, if if we're
Not so practical. Smith then had a new problem to solve, a geometry problem. How to represent infinity on a finite chart? Smith realized he was a bit out of his depth, so he called in the help of two of his colleagues who specialized in mathematics, Ferrell and McRae
Multiply by j again, and by definition you get negative one. It stretched off to infinity , so it wouldn't fit in one chart, and it changed as you moved along the line, so there was no single value for engineers to match.
But what about the integers? 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,
These omega numbers are not bigger than infinity . They just come after infinity . They don't tell us how many things are there, but they do tell us their order.
And that's a remarkable property for a notion of infinity to have, I suppose. But if you thought that there was only one kind of infinity , then it wouldn't be surprising at all, because if you take two infinite sets and put them together, then it's still infinite. And so if there were only one kind of infinity , then it shouldn't be surprising...
- So let's talk about the real numbers. What are the real numbers? Why do they break infinity ? The countable infinity . Looking it up on Perplexity, real numbers include all the numbers that can be represented on the number line, encompassing both rational and irrational numbers. We've spoken about the rational numbers, and the rational numbers, by the way, are by definition, the numbers that can be represented
I don't have these special configurations of water. It's just infinity . - Yeah, infinity absorbs a lot of sins.
know. And according to the algorithm, you're someone that watches our show, but you haven't yet hit that button. Just because infinity is kind of hard for the human brain to wrap your head around, doesn't mean the universe can't
They had collaborated on the Declaration of Independence, and Adams went up to Braintree, Massachusetts And that became the "Infinity War." Oh, did it?
I can divide my token in as many as I want, because it's infinitely dividable-- I don't know how you say. You can infinity divide it-- and then transfer this to anyone, globally, in the world without any overhead or expense. Am I right in understanding this?
I guess the infinity is what makes contents special.
I need an infinity of numbers scattered across all of space.
since Marvel's "Infinity War." Yeah, I mean, if you made-- if you made up-- so first, I've just the simple story, which is--
Having access to infinity is the way in which impulses arise.
It is perfection times infinity .
If I track infinity experiments, I can't possibly deal with it, right?
Interviewer: Almost infinity , infinity hummus.
with infinity but u uh at Gaus himself had admonition all mathematicians treated infinity sort of you know threw long sticks. Okay, they never touched it. Okay, they uh infinity was something that you write under the limit sign. Okay, so you know and you don't think about it anymore and
fact he came up with those wonderful theorems that essentially say that that there are many kinds of infinity . There is the vanilla infinity of the whole numbers and then there is the greater infinity of the continuum and then there are even greater infinities. Okay. And this came as a tremendous shock for mathematicians. They could not refute
He can count to infinity twice. I mean, it's like -- You know, it's ridiculous, right? Like behind his beard is a fist.
Multiply by j again, and by definition you get negative one. Impedances that once ran off to infinity now sit inside this single circle, the entire infinite range captured on one finite chart.
while the node's current cost is infinity .
Dijkstra could update this neighbor from infinity down to six, a total path length of five plus one, but he couldn't update this one.
But as N goes to infinity , we just don't know all of the primes.
We could start with zero and then have one, two, three, all the way to positive infinity and then negative one, negative two, negative three, all the way to negative infinity . This is not how we're used to counting, but both of these options fit the definition of a well ordering.
So we just need a new set of numbers that extends past the naturals, call the next number omega, then omega plus one, omega plus two, and so on. These omega numbers are not bigger than infinity . They just come after infinity .
Take that as a limit to infinity and I'll be chilling...
people's brains when they try to think about infinity , I suppose. This is a property of infinity . - It's a property of infinity that sometimes when you add an element to a set, it doesn't get larger. That's what this example shows. But one can go on with Hilbert's Hotel, for example. I mean, maybe the next day, you know, 20 people show up all at once. We can easily do the same trick again, just move
...and each train car has infinitely many seats. And so we have an infinity of infinities of the train passengers together with the current occupants of the hotel, and everybody on the train wants to check in to Hilbert's Hotel. So the manager can, again, of course, send a message up to all the rooms telling every person to double their room number again. And so that will occupy all the
perspective maybe that you're adopting, but it's not true, and that's the profound achievement that Cantor made is proving that the set of real numbers is not a countable infinity . It's a strictly larger infinity , and therefore there's more than one concept of infinity , more than one size of infinity . - So let's talk about the real numbers. What are the real numbers? Why do they break infinity ?
choice is being used and they don't want to use the axiom of choice, so they work out the consequences that's- that are possible without the axiom of choice or with weakened it's a strictly larger infinity than the natural numbers. But Cantor actually proved a- a much more general fact, namely that for any set whatsoever, the power
I don't have these special configurations of water. - I wonder how often using infinity forces us to deviate
I don't have these special configurations of water. - So even if you're working with infinity , it's good to finitize so that you can have some intuition.