This deceptively simple math problem has stumped mathematicians for almost 300 years! 👉 Head to https://hensonshaving.com/veritasium and enter code ‘Veritasium’ for 100 free blades with the purchase of a razor. Make sure to add both the razor and the blades to your cart for the code to take effect.

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▀▀▀
A huge thank you to Steven Strogatz, Alex Kontorovich, Harald Helfgott, Senia Sheydvasser, Jared Duker Lichtman, Roger Heath-Brown, Kevin Ford, Carl-Fredrik Nyberg Brodda, Kaneenika Sinha & Shehzad Hathi for their invaluable expertise and contributions to this video.

Look out for the upcoming @fern-tv video on the tragic life of Srinivasa Ramanujan! A huge thank you to Ruben Katschko, Mathias Hawk, Chris Varga, Elmer and David from fern for working with us!

The full interview with Steven Strogatz is out now, exclusive to Patreon! – https://www.patreon.com/c/veritasium

▀▀▀
0:00 What is Goldbach’s Conjecture?
4:33 Goldbach and Euler
7:03 The Prime Number Theorem
11:16 The Genius of Ramanujan
14:57 The Circle Method
27:31 Proving the Weak Goldbach Conjecture
32:40 Math vs Mao
36:06 Back to Chen Jingrun
37:15 How you can prove the Strong Goldbach Conjecture

▀▀▀
References:
References can be found here – https://ve42.co/GoldbachRefs

Images & Video:
Image and video references can be found here – https://ve42.co/GoldbachVisuals

▀▀▀
Special thanks to our Patreon supporters:
Adam Foreman, Albert Wenger, Alex Porter, Alexander Tamas, Anton Ragin, armedtoe, Balkrishna Heroor, Bertrand Serlet, Blake Byers, Bruce, Charles Ian Norman Venn, Dave Kircher, David Johnston, David Tseng, Evgeny Skvortsov, Garrett Mueller, Gnare, gpoly, iRick, Jeromy Johnson, Jon Jamison, JT, Juan Benet, Keith England, KeyWestr, Kyi, Lee Redden, Marinus Kuivenhoven, Matthias Wrobel, Meekay, meg noah, Michael Krugman, Orlando Bassotto, Paul Peijzel, Richard Sundvall, Robert Oliveira, Sam Lutfi, Tj Steyn, TTST, Ubiquity Ventures, wolfee

▀▀▀
Writers: Casper Mebius & Derek Muller
Additional Writers: Darius Garewal, Callum Cuttle & Aakasha Bagga
Producer & Director: Casper Mebius
Editors: Jack Saxon & Peter Nelson
Camera Operators: Casper Mebius & Derek Muller
Animators: Fabio Albertelli, Emma Wright, Ivy Tello, Allama Nandii, Emma Foster, Mike Radjabov & Andrew Neet
Illustrators: Maria Gusakovich, Jakub Misiek & Caine Esperanza
Assistant Editor: James Stuart
Researchers: Darius Garewal, Geeta Thakur & Gabe Strong
Thumbnail Designers: Ren Hurley & Ben Powell
Production Team: Rob Beasley Spence, Tori Brittain, Casper Mebius & Gregor Čavlović
Advice & Feedback: Gregor Čavlović
Executive Producers: Derek Muller & Casper Mebius

Additional video/photos supplied by Getty Images, Pond5.
Music from Epidemic Sound

Even the GOAT Tom Brady can’t throw a perfect spiral. Here’s why. 🌏 Get exclusive NordVPN deal here ➵ https://NordVPN.com/veritasium It’s risk free with Nord’s 30 day money-back guarantee!✌

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▀▀▀
A huge thank you to Tom Brady for coming and throwing some footballs with us and letting us analyse his throw. You can check out his channel here: https://www.youtube.com/@tombrady

Also thank you to Kevin Bonner and the team at Shadow Lion for helping make this happen!

A big thanks to Daniel Hare and Wilson for all their help collecting and analyzing the data.

A massive thank you to Prof. Philip Ansell, David Attig, Matt Brotnow, Sam Hince and Derrick Wiberg at the University of Illinois at Urbana-Champaign for all their work testing out footballs in their wind tunnel (more of that to come in our second video).

We’d also like to shout out Stewart Bible, Ryan Sayko and the rest of the team at Resolved Analytics for all their work on the CFD models – https://www.resolvedanalytics.com/

Thank you to Mike Lopez and Tom Bliss from the Football Data and Analytics at the NFL for sharing data and insights from across NFL history.

We’re incredibly grateful to Prof. Richard Price and Major Brett Gibbons for their invaluable contributions to this video.

Thank you to Prof. Chad Orzel, Prof. John Eric Goff, Dr Ainissa Ramirez, Prof. Tony Schmitz & Jonathan Ferguson for all their help on the project.

Also thanks to Chris Anstis for his help as a stand-in QB!

▀▀▀
0:00 How Tom Brady Throws a Football
2:51 Why do you spin a football to throw it?
5:36 How does rifling affect a projectile?
9:09 The Turnover Paradox
11:15 How fast can Tom Brady throw?
14:31 Analysing Brady’s Throw
16:38 Gyroscopic Procession
19:47 It’s impossible to throw straight
23:17 Outdoor vs Dome Stadiums
24:31 Why Tom Brady is the GOAT

▀▀▀

References:
References can be found here – https://ve42.co/BradyReferences

Images & Video:
Image and video references can be found here – https://ve42.co/BradyVisuals

▀▀▀
Special thanks to our Patreon supporters:
Adam Foreman, Albert Wenger, Alex Porter, Alexander Tamas, Anton Ragin, armedtoe, Balkrishna Heroor, Bertrand Serlet, Blake Byers, Bruce, Charles Ian Norman Venn, Dave Kircher, David Johnston, David Tseng, Evgeny Skvortsov, Garrett Mueller, Gnare, gpoly, iRick, Jeromy Johnson, Jon Jamison, JT, Juan Benet, Keith England, KeyWestr, Kyi, Lee Redden, Marinus Kuivenhoven, Matthias Wrobel, Meekay, meg noah, Michael Krugman, Orlando Bassotto, Paul Peijzel, Richard Sundvall, Robert Oliveira, Sam Lutfi, Tj Steyn, TTST, Ubiquity Ventures, wolfee

▀▀▀

Producer & Director: Henry van Dyck
Writing: Henry van Dyck & Derek Muller
Editor: Trenton Oliver
Animators: Mike Radjabov & Fabio Albetelli
Illustrator: Jakub Misiek
Researchers: Emilia Gyles & Gabe Strong
Field Producer: Katy Savage
Camera Operators: John Nicholls, Trenton Oliver, Melkwych Pierre-Louis & Scott Wellington
Drone Operators: Nathan Taupez Scinto, Angelo Fiorini
Sound Recordist: Vlady Garcia
Production Assistant: Leah Jiji
Media Manager: Gabriel C. Rodriguez
Assistant Editor: James Stuart
Thumbnail Designers: Ren Hurley & Ben Powell
Production Team: Josh Pitt, Rob Beasley Spence, Tori Brittain & Barbara Abraul
Executive Producers: Derek Muller & Zoe Heron

Additional video/photos supplied by Getty Images
Music from Epidemic Sound

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I worked at Tesla starting from 2017 thru 2021. I spent most of my time in the distribution and supply chain organizations in leadership positions.

Before Tesla, I was a Director of Business Intelligence and Pricing at the largest Pet Food & Supply distributor in the US, Phillips Pet Food & Supplies. My wife and I also owned a small business in Bethlehem, PA between 2016 and 2019.

I have been a shareholder of Tesla since 2012 and currently own Tesla stock. Nothing I say constitutes as investment or financial advice.

My thoughts are my own and are not representative of everyone who currently works, or has worked at Tesla, Phillips, or any other company or organization I discuss on this channel.

If you like this content I would greatly appreciate your likes and shares!

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Uber’s Future in the Age of Autonomous Vehicles: An Insightful Discussion with The Rideshare Guy

In this episode, we dive into the future of the ride-hailing industry with Harry Campo, also known as The Rideshare Guy. We discuss the impact of autonomous vehicles on companies like Uber and Lyft, and evaluate the potential roles of Waymo and Tesla in this rapidly evolving market. Harry shares his extensive experience from starting as an Uber driver to becoming a significant voice in the rideshare and gig economy. Topics covered include the practical challenges of fleet operations, the importance of driverless technology, and the potential for Tesla’s Robotaxi network to disrupt the current landscape. Tune in for a deep dive into the intersection of rideshare services and autonomous vehicle technology.

00:00 Introduction: The Future of Ride-Sharing
00:39 Meet Harry Campbell: The Ride-Share Expert
01:25 Harry’s Journey: From Aerospace Engineer to Ride-Share Guru
02:43 The Evolution of Ride-Sharing and Autonomy
03:30 Waymo vs. Uber: The Current Landscape
05:01 Tesla’s Role in the Ride-Share Revolution
06:21 Challenges and Opportunities in Autonomous Ride-Sharing
17:51 Insurance and Liability in the Age of Robo-Taxis
33:18 Understanding Peak Demand in Ride-Sharing
34:02 Customizing Fleets for High-Earning Trips
34:58 Nighttime Deliveries and New Market Opportunities
36:33 Tesla’s Advantage in Scaling Robo-Taxis
37:27 Fleet Ownership Models and Their Challenges
39:04 The Role of Tele-Operators in Autonomous Vehicles
41:32 Tesla’s Potential Impact on Uber’s Fleet
47:43 Fragmentation in the Autonomous Vehicle Market
57:04 Uber’s Strategy and Future in the AV Market
01:04:07 Conclusion and Final Thoughts

Now, it’s true that one of the benefits of flying at 30,000 ft is a smoother ride. This is high in the troposphere, the layer in which most weather occurs. So, there’s less turbulence and fewer storms to navigate around.

But this is not the main reason that planes fly so high.

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I worked at Tesla starting from 2017 thru 2021. I spent most of my time in the distribution and supply chain organizations in leadership positions.

Before Tesla, I was a Director of Business Intelligence and Pricing at the largest Pet Food & Supply distributor in the US, Phillips Pet Food & Supplies. My wife and I also owned a small business in Bethlehem, PA between 2016 and 2019.

I have been a shareholder of Tesla since 2012 and currently own Tesla stock. Nothing I say constitutes as investment or financial advice.

My thoughts are my own and are not representative of everyone who currently works, or has worked at Tesla, Phillips, or any other company or organization I discuss on this channel.

If you like this content I would greatly appreciate your likes and shares!

#Tesla #elonmusk #twitter #tsla #TeslaNews #TeslaFSD #TeslaFSDBeta #teslastockanalysis #TeslaUpdate #investing #ElonMuskNews #teslanews #EV #FSD #investing

The biggest chemical cover up in history. PFAS has polluted the entire global water system. Now, potentially dangerous forever chemicals are being found in the entire US population. Go to https://groundnews.com/Ve to get all sides of every story. Subscribe to save 40% off the unlimited Vantage Plan through our link.

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We have a tabletop game launching! Pledge your support now – https://ve42.co/KickStarterD

▀▀▀
0:00 Killed by Fridges
5:27 Teflon and The Manhattan Project
7:59 Teflon is Tricky
11:37 The Teflon Revolution
13:27 Earl Tennant’s Farm
17:34 Inside DuPont
20:28 Fluoride In Drinking Water
25:00 It’s bigger than that
29:23 What is PFAS?
35:56 How much PFAS is in Derek’s blood?
37:56 How forever chemicals get into your blood
46:18 Removing PFAS from drinking water
49:30 Can you lower your PFAS levels?

▀▀▀
A huge thank you to Rob Bilott for his time and expertise. Check out his fantastic book: Bilott, R. (2019). Exposure. Simon and Schuster – https://ve42.co/7R

Rob’s story also inspired the 2019 film: Dark Waters.

Thank you to Doctor Mike for giving us a medical perspective on PFAS! Check him out at @DoctorMike

Thank you to Henrik Haggeman and the Puraffinity team, as well as Andrew Patterson and Eurofins, for doing the PFAS testing.

Thank you to Leslie Hamilton, Johns Hopkins APL, Alex Conrad, Imperial College London, Jana Avgustini, and Matija Krvavica for their help on the project.

▀▀▀
PFAS Tools:
PFAS Blood Test – https://ve42.co/PFASTest
What’s My Exposure? via www-pfas.pfas-exchange.org – https://ve42.co/myexposure
Serum PFAS Calculator via ics.uci.edu – https://ve42.co/pfascalculator
Interactive Map: PFAS Contamination Crisis via http://www.ewg.org – https://ve42.co/pfasmapinteractive
The Map of Forever Pollution via foreverpollution.eu – https://ve42.co/foreverpollutionmap
Australian PFAS Chemicals Map via pfas.australianmap.net – https://ve42.co/australianpfasmap

▀▀▀
References:

References can be found here – https://ve42.co/PFASReferences

Images & Video:

Image and video references can be found here – https://ve42.co/PFASVisuals

3D Models:

“Low Poly Base Mesh-530 Tri” (https://skfb.ly/onsxE) by Huge_Man
“Blood Vessel” (https://skfb.ly/owHJX) by IMU University
“Generic Factory” (https://skfb.ly/onQpR) by assetfactory
“Lowpoly People + Waldo” (https://skfb.ly/6zqCE) by Loïc Norgeot
PFOA Molecule via PubChem – https://pubchem.ncbi.nlm.nih.gov/compound/Perfluorooctanoic-acid

▀▀▀
Special thanks to our Patreon supporters:
Adam Foreman, Albert Wenger, Alex Porter, Alexander Tamas, Anton Ragin, Balkrishna Heroor, Bertrand Serlet, Blake Byers, Bruce, Dave Kircher, David Johnston, David Tseng, Evgeny Skvortsov, Garrett Mueller, Gnare, gpoly, iRick, Jon Jamison, Juan Benet, Keith England, KeyWestr, Kyi, Lee Redden, Marinus Kuivenhoven, Matthias Wrobel, Meekay, meg noah, Michael Krugman, Orlando Bassotto, Paul Peijzel, Richard Sundvall, Sam Lutfi, Samuel White, TTST, Tj Steyn, Ubiquity Ventures, wolfee

▀▀▀
Writers: Gregor Čavlović & Derek Muller
Producer & Director: Gregor Čavlović
Editors: Peter Nelson & Jack Saxon
Camera Operators: Tas Underwood, Emilia Gyles, Gregor Čavlović & Derek Muller
Animators: Andrew Neet, Emma Wright & Fabio Albertelli
Illustrators: Caine Esperanza, Jakub Misiek & Maria Gusakovich
Assistant Editor: James Stuart
Additional Editor: James Horsley
Researchers: Geeta Thakur, Darius Garewal, Gabe Strong & Emilia Gyles
Thumbnail Designers: Ren Hurley & Ben Powell
Production Team: Rob Beasley Spence, Tori Brittain, Casper Mebius, Sulli Yost & Matthew Cavanagh
Executive Producers: Derek Muller & Zoe Heron

Additional video/photos supplied by Getty Images, Pond5.
Music from Epidemic Sound

How a single phone call from a student helped uncover a flaw that nearly toppled Citicorp. Get an exclusive 15% discount on Saily data plans! Use code veritasium at checkout. Download Saily app or go to https://saily.com/veritasium

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Correction: From construction images of Citicorp, sharp-eyed viewers might see that the mid-V columns are still there. Those columns help prevent buckling of the diagonal and the weight of people on the floor above, but do not take the majority of the gravity load. So we left them out of our diagram to clearly explain the load paths. See page 70 of this source for more details – https://drive.google.com/file/d/1ODFo66-_V7Y9vKANbj53mQ0z20RZ2Uai/view

Sign up to the Veritasium newsletter for weekly science updates – https://ve42.co/Newsletter

▀▀▀
0:00 Why is the citicorp building on stilts?
4:44 How wind load works
7:49 Tuned Mass Dampers
11:17 The Anonymous Student
14:00 Quartering Winds
18:38 What were the odds of collapse?
20:31 How was the citicorp building fixed?
25:40 Hurricane Ella
27:00 TMDs Take Over The World
28:36 Conspiracies and Cover Ups

▀▀▀
A big thank you for the incredible people without whom we couldn’t have done this! 🖤

To Michael Greenburg, author of The Great Miscalculation, for his insights that really brought the story and its context to life. He gave us a pre-copy of his upcoming book, and it is fantastic: https://ve42.co/Miscalc
To Eric Hines for all his help with the technical details, fact-checking, and making sure we got the engineering right.
To Michael J. Vardaro for helping us uncover the details of the story and the resources he provided.
To Brock Hedegaard for helping us dive deep into the Citicorp structural analysis. Learn the fundamentals of structural analysis on his brilliant channel, @StructuresProfH, and see here for his analysis of the citicorp center; https://www.youtube.com/watch?v=lE_DspFE2ZM
To Michele Barbato from UC Davis for his inputs on the load calculations.
https://faculty.engineering.ucdavis.edu/barbato/
Matt from @mvvvc for talking us through his wind load analysis.
To Prof. Chris Letchford who helped us understand the impact of wind on Citicorp’s structure.
To DongHun Yeo from NIST for discussing his research about wind effects on buildings similar to citicorp: https://ve42.co/WindLoad
Finally, to Team Columbia —who built the epic TMD and ran wind tunnel experiments that made this story come alive.
This includes the folks at the Mechanical Engineering Teaching Laboratory
(Follow all the cool stuff they’re doing: @mechcu, @columbiaengineering or visit https://teachinglab.me/)
Amanda Lombardo, Manager of Instructional Laboratories
Daniela Durón García, Assistant Manager of Instructional Laboratories
Jorge Casas, Machinist
And the amazing seniors from the Columbia Space Initiative:
(Support their awesome work here: columbiaspace.org/support-us/, Follow them: @columbiaspaceinitiative)
Michael Sheehan
Claudio Solano
Jessica Bolar
Kathryn Lampo

▀▀▀
References:
Michael M. Greenburg. The Great Miscalculation via. nyupress.org – https://ve42.co/Miscalc
Joe Morgenstern. The Fifty-Nine-Story Crisis via. The New Yorker – https://ve42.co/59Crisis
National Academy of Engineering. William LeMessurier-The Fifty-Nine-Story Crisis: A Lesson in Professional Behavior – https://www.youtube.com/watch?v=um-7IlAdAtg
Diane Lee Hartley. Implications of a Major Urban Office Complex: The Scientific, Social and Symbolic Meanings of Citicorp Center via. princeton.edu – https://ve42.co/CitiSymbol
Addition References here: https://ve42.co/citiref

▀▀▀
Special thanks to our Patreon supporters:
Adam Foreman, Albert Wenger, Alex Porter, Alexander Tamas, Anton Ragin, Autodidactic Studios, Balkrishna Heroor, Bertrand Serlet, Blake Byers, Bruce, Dave Kircher, David Johnston, David Tseng, Evgeny Skvortsov, Garrett Mueller, Gnare, gpoly, Greg Scopel, HydrochloRick, Jon Jamison, Juan Benet, Keith England, KeyWestr, Kyi, Lee Redden, Marinus Kuivenhoven, Matthias Wrobel, Meekay, meg noah, Michael Krugman, Orlando Bassotto, Paul Peijzel, Richard Sundvall, Sam Lutfi, Tj Steyn, TTST, Ubiquity Ventures, wolfee

▀▀▀
Written and Directed by Sulli Yost
Edited by Trenton Oliver
Additional Editing by Axel Oliver, James Horsley and Peter Nelson
3D Modeling by Maydelis Anaya Romero
Animation by Mike Radjabov, Andrew Neet, Fabio Albertelli, Jakub Misiek, and Emma Wright
Illustration by Jakub Misiek, Maria Gusakovich and Cainejan Esperanza
Filmed by Sulli Yost, Henry van Dyck, Andrew Abballe, Derek Muller
Produced by Sulli Yost, Derek Muller, Casper Mebius, Zoe Heron, Rob Beasley Spence and Tori Brittain
Additional research by Geeta Thakur
Thumbnail contributions by Ren Hurley and Ben Powell
Additional video/photos supplied by Getty Images, Pond5, Axiom Images and Storyblocks
Music from Epidemic Sound

Why does energy disappear in General Relativity? 👉 Use code VERITASIUM to get 50% off your first monthly KiwiCo Crate! https://www.kiwico.com/VERITASIUM

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▀▀▀
A huge thank you to Prof. Geraint Lewis, Prof. Melissa Franklin, Prof. David Kaiser, Elba Alonso-Monsalve, Richard Behiel, Prof. Leo Stein and Jakob Schwintenberg for their invaluable expertise and contributions to this video.

Special thanks to Alessandro from ScienceClic for his help on animations and input on the script. Check out his excellent video on Noether’s theorem here: https://www.youtube.com/watch?v=hF_uHfSoOGA

▀▀▀
0:00 What is symmetry?
4:25 Emmy Noether and Einstein
7:33 General Covariance
11:59 The Principle of Least Action
15:29 Noether’s First Theorem
18:24 The Continuity Equation
23:20 Escape from Germany
24:49 The Standard Model – Higgs and Quarks

▀▀▀
References:
MathHistory Biography – Emmy Amalie Noether via mathshistory.st-andrews.ac.uk – https://ve42.co/BioNoether
Bartlett, J. G. (2021). The Standard Cosmological Model and CMB Anisotropies. New Astron.Rev. 43 (1999) 83-109 – https://ve42.co/CosmoCMB
Kimberling, C. (1982). Emmy Noether, Greatest Woman Mathematician. Mathematics Teacher, March 1982, Volume 84, Number 3, pp. 246–249. – https://ve42.co/NoetherGreat
Blunck, A., et al. (2016). Mathematik und Gender. Verlag Franzbecker – https://ve42.co/MathGender
Evans, R. J. (1976). The Feminist movement in Germany, 1894-1933. Sage Publications – https://ve42.co/FeministGer
Without Emmy Noether, there would be a huge gap in mathematics and its understanding via http://www.mpg.de – https://ve42.co/NoetherGap
Dihedral group of order 6 via Wikipedia – https://ve42.co/Dihedral6
Emmy Noether University of Erlangen via Wikipedia – https://ve42.co/Erlangen
Timeline of women in mathematics via Wikipedia – https://ve42.co/WomenMath
Elizaveta Litvinova via Wikipedia – https://ve42.co/LitvinovaE
Sofya Kovalevskaya via Wikipedia – https://ve42.co/KovalevskayaS
Charlotte Scott via Wikipedia – https://ve42.co/ScottC
Cornelia Fabri via Wikipedia – https://ve42.co/FabriC
Emmy Noether, the renowned mathematician who taught 20 years of unpaid classes just because she was a woman via english.elpais.com – https://ve42.co/NoetherPay
Noether, E. (1908). Über die Bildung des Formensystems der ternären biquadratischen Form. Reimer – https://ve42.co/NDissert
Noether, E. (1983). Invariante Variationsprobleme. Gott.Nachr.1918:235-257,1918; Transp.Theory Statist.Phys.1:186-207,1971 – https://ve42.co/InvarProb
Noether, E. (2018). M. A. Tavel’s English translation of “Invariante Variationsprobleme”. arXiv – https://ve42.co/InvarProbEn
Byers, N. (1998). E. Noether’s Discovery of the Deep Connection Between Symmetries and Conservation Laws*. Mod.Phys.Lett. A14 (1999) 99-104 – https://ve42.co/SymCon
De Haro, S. (2021). Noether’s Theorems and Energy in General Relativity. Cambridge University Press, 2021 – https://ve42.co/EnergyGR
Rowe, D. E. (2019). Emmy Noether on Energy Conservation in General Relativity. arXiv – https://ve42.co/EnergyCon
Branding, K. (2005). A Note on General Relativity, Energy Conservation, and Noether’s Theorems. Springer – https://ve42.co/GRanEC

Additional References:
https://ve42.co/NoethersAdRefs

▀▀▀
Special thanks to our Patreon supporters:
Adam Foreman, Albert Wenger, Alex Porter, Alexander Tamas, Anton Ragin, Autodidactic Studios, Balkrishna Heroor, Bertrand Serlet, Blake Byers, Bruce, Dave Kircher, David Johnston, David Tseng, Evgeny Skvortsov, Garrett Mueller, Gnare, gpoly, Greg Scopel, iRick, Jon Jamison, Juan Benet, Keith England, KeyWestr, Kyi, Lee Redden, Marinus Kuivenhoven, Matthias Wrobel, Meekay, meg noah, Michael Krugman, Orlando Bassotto, Paul Peijzel, Richard Sundvall, Sam Lutfi, TTST, Tj Steyn, Ubiquity Ventures, wolfee

▀▀▀
Directed by Matthew Davies
Written by Matthew Davies, Casper Mebius, Derek Muller and Pablo Rovetto
Edited by Trenton Oliver, Peter Nelson and Nick Lear
Animated by Fabio Albertelli, Emma Wright, Andrew Neet, Alessandro Roussel and Ivy Tello
Illustrations by Jakub Misiek, Cainejan Esperanza, Tommy Stevens, Maria Gusakovich and Emma Wright
Additional research by Henry van Dyck, Gabe Strong and Darius Garewal
Produced by Matthew Davies, Casper Mebius, Derek Muller, Zoe Heron, Rob Beasley Spence and Tori Brittain

Thumbnail contributions by Jakub Misiek, Ren Hurley, Ben Powell and Peter Sheppard

Additional video/photos supplied by Getty Images and Pond5
Music from Epidemic Sound

How an SAT question became a mathematical paradox.

How do you make infinite choices? 👉 To try everything Brilliant has to offer for free for a full 30 days, visit http://brilliant.org/veritasium. You’ll also get 20% off an annual premium subscription.

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Sign up to the Veritasium newsletter for weekly science updates – https://ve42.co/Newsletter

▀▀▀
A huge thank you to Dr Asaf Karagila, Prof. Alex Kontorovich, Prof. Joel David Hamkins, Prof. Andrew Marks, Prof. Gabriel Goldberg and Prof. Elliot Glazer for their invaluable expertise and contributions to this video.

Head over and sign up to our Patreon for some exclusive behind the scenes footage, showing how the animations and illustrations for this video were made – https://www.patreon.com/posts/patreon-behind-126443880?utm_medium=clipboard_copy&utm_source=copyLink&utm_campaign=postshare_creator&utm_content=join_link

▀▀▀
0:00 What comes after one?
2:42 Some infinities are bigger than others
6:17 The Well Ordering Principle
10:32 Zermelo And The Axiom Of Choice
17:22 Why is the axiom of choice controversial?
23:16 The Banach–Tarski Paradox
27:53 Obviously True, Obviously False
29:58 Your Proof Your Choice

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References:

Up and Atom – https://www.youtube.com/watch?v=X56zst79Xjg
Minutephysics – https://www.youtube.com/watch?v=A-QoutHCu4o
PBS Infinite Series – https://www.youtube.com/watch?v=hcRZadc5KpI&t=125s
Vsauce – https://www.youtube.com/watch?v=s86-Z-CbaHA&t=474s
Ernst Zermelo via Wikipedia – https://ve42.co/zermeloBio
Axiom of choice via Wikipedia – https://ve42.co/choiceAxiom
Georg Cantor via Wikipedia – https://ve42.co/cantorMath
Gregory H. Moore (2013). Consequences of the Axiom of Choice. Dover Publications – https://ve42.co/choiceBook
Georg Cantor (1874). On a property of the class of all real algebraic numbers. Journal für die Reine und Angewandte Mathematik – https://ve42.co/MeyerCantor1874
Heinz-Dieter Ebbinghaus (Dec 2012). Zermelo and the Heidelberg Congress 1904. Historia Mathematica – https://ve42.co/SciDirect1904
Herbert B. Enderton (1977). Elements of Set Theory. – https://ve42.co/SciDirectGCH

Additional References – https://ve42.co/AoCAdRefs

Images & Video:

Foundations of a general theory of sets by Georg Cantor via ViaLibri – https://ve42.co/grundlagen
Alfred Tarski by George Bergman via Wikimedia Commons – https://ve42.co/tarski
Alfred Tarski Offprint Group by Alfred Tarski via Bonhams – https://ve42.co/tarskipaper
La mission strasbourgeoise de Maurice Fréchet by Laurent Mazliak via Images des mathematiques – https://ve42.co/frechet
Kurt Gödel by Alfred Eisenstaedt via New Yorker – https://ve42.co/godel
Leopold Kronecker by Granger via Fine Art America – https://ve42.co/kronecker
Lashi Bandara (2006). Zermelo-Frankel Set Theory and Well Orderings. ResearchGate – https://ve42.co/zermelofrankel
Heidelberg, Germany 1936 by Wagner & Debes via Ward Maps – https://ve42.co/heidelberg
Pythagoras by J. Augustus Knapp via the marginalian – https://ve42.co/pythag
Paul Cohen by C. J. Mozzochi via C. J. Mozzochi – https://ve42.co/paulcohen
Instituto de Matemática Pura e Aplicada. Lecture 01: Introduction: a non-measurable set via Youtube – https://www.youtube.com/watch?v=llnNaRzuvd4&t=834s
Simons Foundation. Fields Medal: James Maynard. Youtube https://www.youtube.com/watch?v=un-z8kgOrV0&t=8s

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