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luv u

Fantastic video! Thanks again for the positively-biased visuals (black-on-white) 🤗

Clicked on this video bc thumbnail stutters when you scroll on your phone.

These are good videos but they always seem like the end way too early. My advice is to wrap everything up by applying what you have said to what you have shown in the intro. I have no idea what a sigma algebra has to do with the volume of a cube

Edit: Sorry, I missed the word 'disjoint'. My bad. Good explanation! It does not seem to me that the Dirac measure is a measure on the power set of X. One of the properties that you give for a measure is that the measure of the union of a collection of subsets is the sum of the measures of the sets. Consider the set X = {1,2,3}, and the Dirac measure where the fixed x=1. Now the measure of {1,2} is 1 and the measure of {1,3} is 1. Their union is {1,2,3}, which should be measure 2 by the above property, but is actually 1 by the definition of the Dirac measure.

I see you are still learning. But keep it up. The quality is quite high in this video.

Is this still manim? Looks good

You should make a followup where you explain how this helps us solve the problem with the cube!

GREAAAT VIDEO KEEP IT UP

Great explanation

I don't see the connection of any of the algebra with the cube you show at the beginning

Infinite chocolate bonbon hack?

An absolute masterpiece

Part regarding how finite subsets would make it work feels omitted. Worth making a follow up? I would approach it from non-measurability point of view, probably by looking at Vitali set and extending

ill watch your other videos this was great

The thumbnail of this video dances in really cool way if you scroll it back and forth. Because the details are really minute and are messing with the rastorisation.

Nice visuals! I've always found Measure Theory to be counterintuitive.

This thumbnail is so trippy when i was scrolling it was moving

At the time 1:15 you say that the map (0,2pi) --> R^2 x |-> (cos(x), sin(x)) « Will give you the whole circle back », how does that include the point (1,0) since 0 and 2pi are not to be mapped?

This playlist has been so helpful... do you plan on making more? 🥹

It really is a nonsensical axiom and it should not have been accepted (even if the resulting theory turns out to be consistent).

Does this mean, you can reach any point from any other point?

So, which part of the sphere is non-measurable? And why? Like, I can totally cut up an orange and real life and I can definitely always measure every piece I cut out.

A surface is not an infinite collection of points. A line is not an infinite collection of points. You can see this because a surface has an area but the area of a point is zero. Zero times infinity is still just zero.

I'm having trouble understanding the second criterion. Could you clarify what you mean by "orbit"?

Did he really need a million dollars? He’s right next to Pythagoras in history. Let that sink in.

And yet people still accept ZFC. So sad that people are obsessed with an axiom set that induces paradoxical results.

Lottery

Great video and awesome animations! Subscribed!

Maybe pointed out already but there is a y missing in the formula of the thumbnail

Set theory is beautiful 🚪

Mandelbrot powers

The paradox is clearly the result of poor axioms. This is inevitable when mathematicians and scientists stay too far removed from the real world (see the almost complete stalling of physics in the last 50 years)

If geometry started from 0D instead of 1D there wouldn't be a Banach-Tarski paradox. Euclid got Plato's "forms" and "solids" upside-down like 2300 years ago and we still haven't fixed it.

HoTT vs ZFC

1:52 is there at the first point a "(" missing?

A nobel prize isn't the most prestigious science award. It's just the most popular one. Clearly if experimentalists won the prize based on his predictions the he would have done so if proven within his life time. The thing here is that bell should be immortalized as one of the more prominent figures in quantum theory. The prize is irrelevant and has a lot of politics around it.

I amazed myself that I could read that thumbnail math notation.

The full Axiom of Choice is not needed, you can prove the Banach-Tarski Paradox using just the Hahn-Banach theorem, ( ZF+Hahn-Banach). You are going to want Hahn-Banach around for functional analysis and operator algebras, which provide the mathematical foundation of quantum mechanics. It is worth noting that a key object used by the Banach-Tarski Paradox construction is the existence of a free group of rotations with two generators, which requires irrational rotations. The Banach-Tarski paradox is a consequence of the existence of irrational numbers.

Just how is the formula from the thumbnail even syntactically correct?

You don't outright say it, but you seem to be implying that cantor's diagonalization relies on axiom of choice. This is not the case.

I have to say "There is always some way to choose a point", together with the formal way of axiom of choice, is something that does not make any sense to a newbee if you don't illustrate what "way" and "choose" and "A" and "y" mean.

I wouldn't describe Georg Cantor as “Russian-born” not because it's false (he was born in St. Petersburg), but because it's misleading. Cantor was German. His name is German and he published in German, and lived most of his life in Germany.

Technically speaking, to prove Banach-Tarski, you don't need the Foundation Axiom.

At the start of this video and the last one you describe the Banach-Tarski paradox as only requiring splitting the sphere into finitely many parts, but this is not true. I’m guessing you meant infinitely many points?

Shouldn’t you have to measure the surface area of the sphere? The sphere is only the points equidistant from the center, not the points inside the surface

This channel is a hidden gem. This is some of the most thoughtful and well made content I have seen in literal years. Here’s to many more great videos!

Nice video. However, while there's nothing wrong with the conclusions, there are some quite imprecise statements in it (e.g. "at the end of this infinite process" when talking about Cantor's Diagonalization). You don't need to finish the infinite diagonalization, just show that for any member of the set, the diagonal isn't going to match.

There are only two kinds of things: The ones that can be grouped And the ones that can be grouped in a more esoteric way