Support Vsauce, your brain, Alzheimer's research, and other YouTube educators by joining THE CURIOSITY BOX: a seasonal delivery of viral science toys made by Vsauce! A portion of all proceeds goes to Alzheimer's research and our Inquisitive Fellowship, a program that gives money and resources directly to growing science channels here on YouTube! null my twitter: @tweetsauce my instagram: electricpants Sources and links to learn more below! I’m very grateful to mathematician Hugh Woodin, Professor of Philosophy and Mathematics at Harvard, for taking the time on multiple occasions to discuss this topic with me and help me wrap my (finite) head around it. I’m also grateful to David Eisenbud, the Director of the Mathematical Sciences Research Institute (MSRI) and professor of mathematics at the University of California, Berkeley, for his help and for connecting me with Hugh Woodin. And of course, big thanks to Brady Haran who created the “mile of pi” seen in this video and connected me with all these mathematicians in the first place. His channel, Numberphile, is superb: null BOOKS related to these topics that I used: “The Outer Limits of Reason” by Noson S. Yanofsky: http://amzn.com/0262019353 “Infinity and The Mind” by Rudy Rucker: http://amzn.com/0691121273 “Roads to Infinity” by John C. Stilwell: http://amzn.com/1568814666 “More Precisely: The Math You Need to Do Philosophy” by Eric Steinhart: http://amzn.com/1551119099 “Satan, Cantor and Infinity: Mind-Boggling Puzzles” by Raymond M. Smullyan: http://amzn.com/0486470369 classic book that helps introduce concept of axioms: “Introduction to the Foundations of Mathematics” by Raymond L. Wilder: http://amzn.com/0486488209 Hugh Woodin speaking about infinity at the World Science Festival: https://www.youtube.com/watch?v=KDCJZ81PwVM Names of large (finite) numbers: https://en.wikipedia.org/wiki/Names_of_large_numbers Geoglyphs: The biggest number: https://goo.gl/maps/7GWcpnzo7iG2?entry=yt Fovant badges: https://en.wikipedia.org/wiki/Fovant_Badges Battalion Park: https://en.wikipedia.org/wiki/Battalion_Park A mile of pi [VIDEO]: https://www.youtube.com/watch?v=0r3cEKZiLmg Wikipedia’s great visualization of ordinals out to omega^omega: https://en.wikipedia.org/wiki/File:Omega-exp-omega-labeled.svg as seen on: https://en.wikipedia.org/wiki/Ordinal_number this is also a good page about ordinals: http://math.wikia.com/wiki/Ordinal_Number also: http://www.cut-the-knot.org/WhatIs/Infinity/Ordinals.shtml and: https://en.wikipedia.org/wiki/Order_type and https://en.wikipedia.org/wiki/Well-order Axioms: https://en.wikipedia.org/wiki/Axiom https://www.reddit.com/r/askscience/comments/30xokb/have_there_been_axioms_that_later_have_been/ http://philosophy.stackexchange.com/questions/5922/what-is-the-difference-between-dogma-and-an-axiom http://www.cs.umd.edu/~gasarch/BLOGPAPERS/belaxioms1.pdf http://www.cs.umd.edu/~gasarch/BLOGPAPERS/belaxioms2.pdf THE UNREASONABLE EFFECTIVENSS OF MATHEMATICS IN THE NATURAL SCIENCES [PDF]: http://www.maths.ed.ac.uk/~aar/papers/wigner.pdf https://en.wikipedia.org/wiki/The_Unreasonable_Effectiveness_of_Mathematics_in_the_Natural_Sciences Large Cardinal game based on 2048: http://cantorontheshore.blogspot.it/2014/10/one-reinhardt-and-counting.html Other good resources: http://quibb.blogspot.com/2012/01/infinity-first-transfinite-cardinal.html http://plato.stanford.edu/entries/set-theory/ http://cantorsattic.info/Cantor%27s_Attic http://cantorontheshore.blogspot.co.at/2014/09/artemis-fowl-and-large-cardinals_22.html http://isomorphism.es/post/10782081422/what-comes-after-infinity-transfinite-arithmetic-and-ord https://lukepalmer.wordpress.com/2007/06/14/the-lesser-of-infinitely-many-evils/ MUSIC BY: null http://www.audionetwork.com/