How the Axiom of Choice Gives Sizeless Sets | Infinite Series

Does every set - or collection of numbers - have a size: a length or a width? In other words, is it possible for a set to be sizeless? This in an updated version of our September 8th video. We found an error in our previous video and corrected it within this version.

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Previous Episodes
Your Brain as Math - Part 1

Simplicial Complexes - Your Brain as Math Part 2

Your Mind Is Eight-Dimensional - Your Brain as Math Part 3

In this episode, we look at creating sizeless sets which we call size the Lebesgue measure - it formalizes the notion of length in one dimension, area in two dimensions and volume in three dimensions.

Written and Hosted by Kelsey Houston-Edwards
Produced by Rusty Ward
Graphics by Ray Lux
Assistant Editing and Sound Design by Mike Petrow

Resources:

Vsauce

Special Thanks: Lian Smythe and James Barnes

Thanks to Mauricio Pacheco and Nicholas Rose who are supporting us at the Lemma level on Patreon!

And thanks to Matthew O'Connor and Yana Chernobilsky who are supporting us at the Theorem Level on Patreon!

[veggiet]

boy I miss "Infinite Series..." I wish PBS would bring it back

[RolandHutchinson]

"Pretty much any set that you can think of has a size." (It's the sets that you can't think of that get you.)

[Desrathedemon]

To be contrary, my favorite consequence of the axiom of choice is that it allows us to compare the sizes of sets. Given two sets α and β, either |α| ≤ |β| or |β| ≤ |α|. This is the notion of 𝘤𝘢𝘳𝘥𝘪𝘯𝘢𝘭 𝘤𝘰𝘮𝘱𝘢𝘳𝘢𝘣𝘪𝘭𝘪𝘵𝘺, and it's equivalent to the axiom of choice.

[conduit]

“Let us know in the comments your favorite consequence of the Axiom of Choice”

The single greatest call to action in the history of YouTube 😍

[lchtrmn]

On the whole, this is quite good. However, I don't believe any credit is ascribed to the mathematician that discovered this set, G. Vitali. This measureless set constructed by the application of the Axiom of Choice to the equivalent classes is known as the Vitali Construction or Vitali Set.

[DontMockMySmock]

My favorite consequence of the Axiom of Choice is mathematicians flipping their shit

[ObjectsInMotion]

Wanna see an anagram of Banach Tarski?


Banach Tarski Banach Tarski

[harryandruschak]

"The Axiom of Choice is obviously true, the well-ordering principle obviously false, and who can tell about Zorn's lemma?"

[Hedning]

3:35 it took me 5 minutes to realize you must mean "each" rather than "all". "all numbers show up in 1 bin" can easily be interpreted as "all numbers show up in the same bin".

[curtiswfranks]

Man, do I miss this series. By its very own name, it was a series which should never have ended. :'(

Well, anyway, I am sharing this video with one of my students because it is one of the best explanations that I have ever seen.

[darrellee]

My favorite consequence of the axiom choice is that you chose to make these videos for us. thanks.

[Bodyknock]

This video got me thinking about measures and infinite sets and I think it would be interesting if Infinite Series did a piece on an extended number system such as the surreals or hyperreals that include infinitesimals and what that implies for those systems in terms of if or whether they can support something similar to a Lebesgue measure. (Heck, the surreals are kind of cool in general.)

[tuqann]

Beautiful math presented beautifully. New post on an old video, I hope this channel gets reborn.

[kiledamgaardasmussen]

While it seems reasonable to discard the Axiom of Choice, the consequences of Axiom of No Choice are similarly weird: without choice, there exist vector spaces without a basis; there are two different non-equivalent ways of defining continuous functions; and the cardinality of two sets may not be comparable.

[AndrewRod]

What really alleviates my sense of anxiety over the paradox is your soothing voice and your calm and relaxing way to explain mathematics. Thank you Kelsey, as long as you guide, I have no fear of diving into the math universe.

[jymmy]

My favourite consequence of that axiom is that if we accept it, we can prove that each set admits a well-ordering relation. This means that, each subset of R has a minimum (of course, the relation might not necessarily be the usual less-than-or-equal-to, and in fact, with R it is not!)

[matthewcheung]

My favorite consequence of AC, or rather equivalence, is Zorn’s lemma. I want my rings to have maximal ideals.

[ElPasoJoe]

In the midst of my first year graduate study in Mathematics, one of the professors stated that the purpose was to educate our intuitions. He then produced the Cantor ternary set. Yup. My intuition felt educated...

[RajinderSingh-tnpu]

Just subscribed and got a new math video in the notification bar.Thanks for making such an amazing video

[shaheerziya]

What a well produced and great series. Such a shame that they stopped.