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 diagonal class, you know, the class of all sets that are not elements of themselves. If that were a set, then it would be an
number Z never had any zeros or nines in it, it wasn't one of those numbers. And so actually, in those cases, we didn't need to do anything special to diagonalize. Just the mere fact that our number has a unique representation already means that it's not equal to those numbers. So maybe it was controversial in Cantor's day more than 100 years ago, but I think it's most commonly looked at today as, you know, one of the
and it was snowing at all times it was gray very very gray very little contrast and on the day of the coronation there were three diagonal white lines across the screen but you could just discern plopping horse feet the odd wheel some coronet at some point but really none of us i think who watched the
or tension than they are with bending loads. So if we add diagonal bracing, they can carry this horizontal load. The beams sort of act like springs, and when they're compressed, they push on the joints.
You wrote down some axioms, and then you proved some theorems. These other diagonal components, Txx, Tyy, Tzz, these turn out to be the pressure exerted in some kind of fluid.
You wrote down some axioms, and then you proved some theorems. And the off-diagonal are everything else, the momentum, the heat flow, the stress.
name. And in replacing it, they could decide which way um the the city would be anchored. If they anchored the bridge on a diagonal , that would probably become the core grid for the entire city. If they went north south, then it would stay in a rectal linear north south grid. And this is an important enough decision that they put it to a vote. They put let every citizen in the
Ah. It was a brilliant idea. I like diagonal lines, as a lot of you know, but not here. I was going to leave out this one and then cover that end and do it like that and yeah.
Wait. Wait, wait, show me, Daniel. Without the diagonal . OK, so you go diagonally down and you stop here. Right. And then you go up and then you go like that and then you-- Could you do that without using a diagonal ,
And you don't really need to go and get that sugar. Bias just means diagonal .
to where the cranks are and then a very low rear end to the back wheel and then kind of this diagonal member off to the seat so the the question was um how do we address the directionality of using carbon um versus a a material like
But if you identify using, for example, there's a movement called, or there are two movements called the "chop and lift" where you're using, generally speaking, a cable and you're doing this type chopping diagonal movement and then you're reversing it and going up. And so you test your four quadrants and you'll generally find that one quadrant is extremely weak, relatively speaking. And when you fix that it's incredible how many aches and pains,
So those states are pretty persistent. But the off diagonals are unstable. For example, you think about a transition from an extractive to a more inclusive society, South Africa.
It was kind of an Etch-A-Sketch kind of an experience. Because to go diagonally you had to do it in a zigzag. And they received distractive entertainment.
But there's also a, sort of, nicer, more methodical, way of writing this out, and it will help us build some intuition. - We'll start by drawing two diagonal lines. - I'm building like a pyramid.
He grabbed a napkin and sketched out an idea. He drew six layers of diagonal braces up each face of the tower. These chevrons would transfer the forces to the middle of each face and down to the stilts.
Then I called him back and pointed it out to him that there's some peculiar things about this building. The worst loading case was not the diagonal , but it was the ordinary wind that everybody thinks about. The wind pushes straight on the building.
You wrote down some axioms, and then you proved some theorems. It's nice and pretty and diagonal , in this case.
when Yuka-san and my makeup were integrated. The angle is usually our profile or diagonal .
There were dirt roads and there was an area just to the south of the core part of the city that had a road called the old country road that went on a diagonal . And that road was becoming more popular. So some land owners, some some people came and brought bought this property with the thinking eventually this area would grow and become really
And here is the ancestor to the George Washington Bridge, the Brooklyn Bridge, which we're all familiar with. The trademark of the bridge is those diagonal stays. There was a big concern about how the counteracting sway and avoiding the cyclical sinusoidal movements
I don't even need to name the countries, you can figure them out for yourselves. But China is on one diagonal . And indeed, there are some other cultures that I've experienced where people are obedient if you don't observe them too much and they become disobedient
It's a little bit more complicated than that in South Africa, but that's the idea. So what about the other off diagonal ? Well, you could think of China, for example.
But basically it's a diagonal more or less, meaning that women think guys who are their same age are the best looking.
I'm over here still below the diagonal ; that's the good place to be. The ones above the diagonals we call losers because --
He computed the forces for each, as we did before, and summed up the result, but then he noticed something strange. Then now we look at the diagonals , the stresses in half of them vanish, and in the other half, double. Since the force on each side was F over the square root of two, these beams get double that.
is actually fairly easy to diagonalize and create something that's reasonably consistent, and clear, and clean in terms of what's the architecture
No. You could also go-- No, I'm sorry. I should have said can't go diagonally . Only up and down and left and right.
of buildings as they would fit together in a plan. And this is showing how all these diagonals are converging on a single point of view. As an architectural photographer, with a view camera, you learn how to actually move that vanishing point around,
And it's often called exact diagonalization or full configuration interaction in chemistry.
But they have to be underpinned by inclusive political institutions. So what about the off diagonals ? Well, what we say about the off diagonals is that the off diagonals are unstable.
There's a grid at right angles. And if you know that you've traveled in some diagonal path, and you know what your x displacement is and your y displacement, Pythagoras' theorem says that d, the distance you've moved, is the hypotenuse of a right triangle.
The office of "The Wall Street Journal" then was diagonal from the World Trade Center.
I've done some where everything's at a diagonal .
If you take it and twist it on the diagonal , twist it 45 degrees, you've got a diagonal .
And that's sort of-- we all know a diagonal or a line or a horizontal bar.
And you can see if you follow the diagonal down to the right, it goes from 70% to 63.46% to 60.10%, 58.09%.
This pizza I kind of always like to go diagonal .
I always think that even my cut will be a diagonal cut.
Actually, diagonal , because that was-- it just so happens to fit.
The place you wanna be is below the diagonal . My buddy Mark Smith's right over here; he gets attended to quite well. At this point, I was only a moderately active Twitter person;
they, because they required diagonal cuts and I didn't have that machine.
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
idea has proved to be an extremely fruitful proof method, and almost every major result in mathematical logic is using in an abstract way the idea of diagonalization. It was really the start of so many other observations that were made, including Russell's paradox and the halting problem and the recursion theorem, and so many other principles are using diagonalization at their core. So...
You might be able to make something work about diagonals if you try hard enough.
digits zero or nine. I'm just always using the other digits and not zero and it would form a kind of diagonal going down and to the right, and that, for that reason, this argument is called the diagonal argument because we're looking at the Nth digit of the Nth number, and those exist on a kind of diagonal going down. And we've made our number Z so that the Nth digit of Z is different from the Nth digit of the Nth number.
central to the the township of Missoula. and he saw this um these these younger gentlemen buying the old this this diagonal road area. He was like, I don't I don't like this violation of the grid system. Uh he thought that maybe to the south they wanted to create South
and uh the way it works is when you spin one motor actually goes diagonal and when you spin the other one it goes the
In fact, if I drew a diagonal that hugged this envelope right here, these are the hardest to find, below that diagonal .
And most significantly of all is this smear at the bottom of the picture-- this diagonal crude smear, which people